Does there exist a chemical-free, homemade substitute to deli?

Roast beef is relatively easy to make at home and is much more delicious that store-bought. Here is a good roast beef recipe from the New York Times.

There is a Japanese method of cooking chicken called "Torihamu" where the goal is to make the chicken the texture and flavor of deli ham. Here is a little more information and a recipe.

A lot of the flavors of deli meat are dependent on salt. A deli meat without salt just does not taste the same. Remember that when making it at home, you will always be using less salt than a manufactured meat. Celery salt may also give you the flavor you need.

There's also a lot of good recipes like this DIY Homemade Lunch Meat.

Here is a video for making deli ham. The trick to that deli ham texture is cooking gently in water and then pressing.

I hope this was helpful! Best of luck to you.

It is certainly possible to make classic deli-meats at home. Most "deli meats" fall into the cured category though which means that the preservative agents are still necessary to prevent food borne illnesses such as botulism. Nitrites and nitrates alongside salt are the main inhibitors in this case. At home you CAN safely leave out most of the "fillers" that are often used in commercially prepared things.

There is FAR too much to detail here regarding meat curing, however there IS a plethora of information available online regarding everything from ham to salami and more. I encourage you to look into something called "Charcuterie" which is the practice/art of making cured meats and sausages (which is what most deli meat is).

https://en.wikipedia.org/wiki/Charcuterie

There are many books about the subject too. Off hand, several books regarding meat curing and butchery are: The Whole Beast, Charcuterie: The Craft Of Salting Smoking And Curing, Home Production of Quality Meats and Sausages.

If you are not interested in curing meats, smoking them may be another option: things such as smoked turkey breast are common deli meats too. Again, there is too much about smoking to list here, however there is SO much information online about it. You can either buy a smoker or make your own (google "DIY smoker"). The smoking process generally causes the meat to loose a lot of moisture in addition to adding the smokey flavour which should transform them into something similar to that found in a deli.

Nutrition is off topic here and this is kind of in between a nutrition and cooking question. You might want to ask a nutrition focused version of this on health.stackexcange.com

I am surprised you are finding packaged lunch meat taste better than what you prepare.

My neighbor just grills a chicken breast for his daughter every school night while he is cooking dinner.

I like to crock chicken thighs. To me they taste better than a breast in a sandwich as a little more moist. And the are 1/2 the price. I do skin side down with no extra oil and some oregano. Garlic and onion if I have it. You can keep them in the fridge for 2-3 days and freeze for a lot longer. I let them cool just bit and then remove the skin and bone. To me it is easier to remove the bone after it is cooked.

I used to trim the fat and skin first but it would be dry. Removing skin last and drain is a lot easier.

Below is actually from the BBQ but same basic idea. BBQ taste better than crock but more prone to overcooking as a pretty narrow window. On the left is skins for my dogs and next bone and straight to the fridge.