How do I get a chewy crust from homemade pizza dough?

In general, you're going to be unable to reproduce pizzaria crust at home oven temperatures. They are most likely baking their pizza at between 600F and 800F for only a few minutes, which produces a crust texture you can't achieve at 400F or even 500F.

This is the reason so many people are trying things like turning backyard grills into ad-hoc pizza ovens.

However, these general tips will help:

1. use a pizza stone and put the pizza on the stone directly
2. heat the oven to 500F for at least 1/2 hour, with the stone in it, before baking
3. roll your pizza dough very thin
4. don't put a lot of wet toppings on it

You also might want to consider cultivating a sourdough pizza crust, which will be chewier than one risen with commercial yeast.

I recommend starting with a chilled dough, puncturing with a knife or fork (see below), and cooking in a greased pan initially then bricking.

Cooking directly on the stone the entire time works better for thinner crusts than mid-dough. By mid-dough, I am referring to the ones that plump up a bit and finish between 1/4" - 1/2" in thickness. They can have both uniform, almost cake-y cross-sections, or air pockets and bubbles depending on how kneaded they are (more kneading means cake-ier, denser dough).

I worked in a shop that made a mid-dough similar to what you describe. It was a standard dough recipe (flour, water, yeast), but I have had success replicating it at home at 450'F with beer and less kneading. At home I use a pan and then finish either on the rack or a stone. In any case, you get a dough that tears nicely and has a good chew.

The dough itself was mixed in the morning and kept refrigerated for as long as 12 hours as individual shells in greased pizza pans covered with saran wrap. To prepare a pizza, we pulled the shell, punched it with a fork (puncture a ring around the edge to create a crust, puncture the inside to allow for air). Dress with sauce and toppings.

We had a stone-bottomed oven, but initially the pizza is baked in the oven in the pan until the bottom of the dough has hardened enough to get "bricked" (the dough should be rigid enough as to be removed from the pan with one spatula). At this point you remove the dough from the pan with a spatula and place it directly on the brick. Cook until the top of the dough browns, and the cheese and other toppings are evenly browned.

I realize I'm coming into this rather late, however you may want to try adding molasses, instead of sugar, to the dough. This will make the crust more chewier/soft instead of crispier. It's what you put in the dough that will determine the end result.

Some additional notes referring to my experience;

• When placing pizza dough in the cooking tray, make the sides of the dough a bit higher by pressing gently to the sides with your fingers in order not to let any topping’s juice to go underneath the dough.
• The cooking tray should be slightly oiled beforehand.
• Preheated oven is a “must”.
• Pizza dough should be softer than bread dough.
• When brushing pizza dough with tomato sauce (done before putting any toppings on) make sure not to use too much sauce.
• I use bottom of the oven or the first grid depending on whether the pizza tray is stoneware or metal.
• When the bottom still needs to be cooked, I place a thick sheet of aluminium foil on top of the pizza to prevent the toppings getting dried out/burned (not covering with foil, just covering it gently).

You need to develop the gluten in the flour to get a chewy consistency. Be sure to knead the dough until is smoothly stretches when you pull it apart.

Another late answer here, but if you want that "raw" dumplingy layer under the sauce, and a crispy bottom, oil the pan (unless you use a stone) and use a sauce like that for spaghetti, instead of a thicker, pasty pizza sauce. I find for me I get that layer and it's my favorite part.

I usually stretch mine into an oiled pan, and bake 12 min exactly in a 450 degree oven and I get crispy golden crust, soft fluffy chewy inner, and that dumpling like layer you're looking for.

Nobody mentioned the flour, I use bread flour for pizza crust and add a bit of gluten too. The gluten and kneading help produce the chewy, elastic crust...

I use a recipe using 1/2 cup semolina flour and 2 1/2 cups bread flour, called Cheri's Favorite Pizza Dough.