How do I cook a chicken to make it really really really soft?

How you prep the chicken is also as important as how you cook it. Cooking method is only half of the answer. Brining the meat in a salty/sugary/acidic solution will go a long way in adding flavor and making the meat moist and tender.

Braise it. Low and slow braising will help make almost any meat fall apart tender (though dark meat is best, thighs/legs). Molly Stevens All About Braising cookbook is a great intro into this technique. For whole chicken braising google search for Jamies milk braised chicken. Makes extremely tender braised chicken and is a great into recipe.

Slow cooker can make any meat really soft.

The best tip I can give is to use chicken thighs rather than breast. They have more fat, which keeps the meat moist. They need slower cooking but are well worth it.

Try a coq au vin style dish - brown the chicken thighs in a little oil, then take them out, fry off some onions, mushrooms and garlic, then add a couple of glasses of red wine, some chicken stock, herbs, and the thighs. Simmer for two and a half hours and the chicken will be falling apart.

If you insist on using chicken breast, try this: stuff the breasts with something reasonably fatty - try feta cheese, sundried tomatoes, garlic and lemon zest. Fry in a hot pan for a couple of minutes either side to brown. Then, take a circle of baking parchment big enough to fit in the pan, scrunch it up, and run it under the cold tap. Shake it out, pop it on top of the chicken, then put a heavy saucepan lid on top and turn the heat down to low-ish. Let it sit under there for 15-20 minutes, and you should find the chicken is nice and tender.

I used to have trouble getting chicken to come out tender, but I've had a lot of luck with a couple of simple rules of thumb.

1. Liberally salt the chicken beforehand and let sit 20-30 minutes.
2. Do your very best not to overcook it. It's a fine line between undercooked (which is dangerous) and overcooked (which is unpleasant to eat), and it's easy to err on the side of unpleasant in order to avoid dangerous. Just tread carefully and you can hit that perfect middle ground pretty reliably.
3. Finish the meat with a convection cooking method, even if it doesn't start that way. I like to pan-sear chicken breast, then finish in the oven. This lets you get the outside you're looking for, while drastically reducing the risk of overcooking (see point 2.)

I've found dry salting to be as effective or more effective as compared to brining, and less trouble. It's more common with steaks, but I do it all the time with chicken. Basically, you liberally salt the chicken, and season it as you see fit, then let it sit for at least 20 - 30 minutes, but no more than an hour.

Forgive my vague science here, but as I understand it, the salt first draws moisture out of the meat, then, once it has drawn out enough moisture to saturate the salt, the meat soaks up the now salted water (along with whatever other seasonings you've included). Once the salt has permeated the meat, it begins to break down the proteins, so they don't have a chance to seize up when heated and become rubbery.

Overcooking is an obvious problem, and one usually caused by paranoia about salmonella, which is an entirely reasonable concern. Don't get me wrong -- you SHOULD worry about salmonella. But don't over-do it. 165F is enough to kill off harmful bacteria; with an interior temp of 165, the chicken is safe to eat. If you cut into it, the meat should be white, not pink, and the juice should run clear. This should be a last-minute double-check though; don't cut into the meat unless you're fairly certain it's done, or you'll let all the juice out and it will come out dry.

And finishing it off using a convection cooking method just makes this a little bit easier. When searing the outside of a cut of meat, whether it's in a pan, on the grill, or whatever, the outside gets very hot before the inside reaches a food-safe temperature. If you continue cooking this way until the inside is cooked through, the outside will be overcooked, and the rest will be somewhere in between done and overdone. The goal, of course, is for just the outermost parts to be seared/charred/etc., and all the rest of the meat, all the way through, to be just done. Convection cooking will get you there much more gently and safely, and make it much easier to get the meat juicy and tender. So, sear in the pan and finish in the oven; or char on the grill on high, then drop the heat to low and close the lid to let it finish cooking through.

Another trick, once you've got a feel for how long it takes a cut of a certain size and thickness to cook through, is to pull it off just a touch early, cover it with aluminum foil, and let the residual heat finish cooking it. Since it's just equalizing its internal temperature, instead of continuing to heat up, this will also help avoid overcooking.

Iranian poultry is braised. A reaction converts the collagen to chicken-flavored gelatin, and the meat yields like butter. I've braised chicken into oblivion, so don't overdo the braise time.

The dish is probably zereshk polo, which is served with barberry and pistachio rice.

A ten step summary:

1. Thighs not breast, bone in, skin off.
2. Salt, pepper and garlic powder.
3. Sear chicken and skin in canola.
4. Remove chicken and skin.
5. Brown onion, then brown garlic.
6. Put chicken and skin back.
7. Cover halfway with lime juice, a spoon of tomato paste, and saffron infused water.
8. Add a pod of cardamom or a pinch of turmeric, and a bit of butter.
9. Cook slow & low.

The braising acid is the tomato and lime.

If the mystery dish was like a sweet/tart mole-like dish, then it was probably fesenjan. It is made differently, with pulverized walnuts and pomegranate juice, but the cooking process is very similar. The walnuts are fried with the spices, the pomegranate (acid) is added to the braise.

Use quality chicken.
I've found that buying quality whole chickens and butchering them myself greatly increases the tenderness and flavor of the chicken. In fact the first time I did this I was convinced I had undercooked the chicken despite what my thermometer told me. You are going to have a hard time creating melt-in-your-mouth tender chicken using flash-frozen 12 for $10 chicken breasts using any technique.

I would recommend poaching in water or any type of liquid: wine, water, chicken broth etc. This is even good for a whole chicken. Add salt and cook.

I have been experimenting with sous vide. I have a cheap temperature controller that controls a rice cooker with a mechanical switch. It's a bit in accurate. The temperature fluctuates between the set temp to +3C. But the results are incredible.

http://www.douglasbaldwin.com/sous-vide.html

http://www.sousvideathome.com.au/

Drawn out thread, but it's a high result in google so I'll answer.

If the Iranian dish you're referring to was seasoned with cardamom and served with basmati rice, barberries and caramelized onions it may have been from a cookbook called "Jerusalem". It's an awesome recipe and following it led to my very first experience with braised chicken that didn't have fatty, rubbery skin. It doesn't come out crisp, but it very soft and just sort of falls apart in your mouth. It's hard to describe but it's really good.

I'm still not sure what the exact trick is, but in the recipe the chicken is seared well for 5 full minutes on each side, and then nestled amongst rice, enough water for the rice and the other ingredients. The liquid does not cover the skin. It's then cooked covered on VERY low heat for 30 minutes. Next, it's taken off the heat, a towel is placed between the pot and lid, and it's allowed to steam under residual heat for 10 minutes.

If you want to get it moist, tender and juicy every time, invest in a sous-vide setup. These used to be expensive to get, but a lot of cheaper models have recently come out.

As others have pointed out, a rubbery texture is caused by overcooking the meat. Brining helps somewhat in helping retain moisture and gives you a wider margin for error, it's still possible to overcook brined meat.

Since the degree of cooking (under-cooked, over-cooked, or perfect) is entirely determined by temperature, you need a way to precisely control it. Sous-vide is the best way I know of to accomplish this.

I cook all of my chicken breast sous-vide at 58.5°C (137°F). Thigh meat requires a higher temperature of around 65°C. Cooking time depends on the thickness of the meat; there are formulas by Doug Baldwin and apps such as Sous Vide Dash to work out how long it will take for the centre of the meat to reach this temperature and pasteurize to kill salmonella and other bugs.

Others pointed out the importance of meat quality, and this is something which definitely becomes apparent under the controlled conditions of sous-vide cooking. I have done a fair few tests, and as a result now strongly prefer organic chicken breast for texture and tenderness. Picking the right supplier means I've been able to eliminate stringy textures or pappy mouthfeel, and end up with chicken that cuts cleanly like a good steak every time. I have not had adverse results using meat I have vacuum packed and frozen myself prior to cooking.

Two words: "slow cooker." Even a stewing chicken will be fall-off-the-bone soft after being slow-cooked.