I've made a couple of deep dish pizzas over the last several months and I occasionally get either a soggy or a not cooked crust. I've been pushing the dough into a cast iron skillet and then putting it into a 400-450 degree oven for say around 25 minutes. By then the crust around the edges is threatening on burning but the crust on the bottom might still be doughy.
I've thought of a couple things and I'd love to hear some feedback on what you may have tried and found successful:
I'm nervous that the cast iron might suck enough heat out of the stone to risk cracking it.
As always your feedback is appreciated.
For a deep-dish pizza, around 425°F is right, and so is 20–30 minutes. That's starting with cold dough (need to keep the butter layers chilled, at least for a Chicago-style pizza).
Cooking in an aluminum 3" deep cake pan is fine. I suppose cast iron should work too (though it'll heat slower, so might take longer). As has been pointed out in comments, the cast iron much greater heat capacity may be part of the problem; I'd guess preheating it would help. (You can just put the cast iron skillet in the oven as the oven heats, though depending on how long you let the oven heat, you may want it in for only part of the time—no idea what the optimal temperature for it is). Make sure to have plenty of oil under the dough, and also cook on a lower oven rack.
I'd guess that you're using too-watery toppings. The tomato sauce should be pretty thick, much thicker than you'd ever use on a thin-crust pizza. Vegetables may need sweating to get some moisture out. If nothing else works, partially cook the crust (say, ten minutes or so) and then add the sauce and toppings.
I can vouch by Cook's Illustrated's Chicago-Style Deep Dish Pizza recipe. Normally they have a paywall, but currently that recipe isn't behind it, so grab it quick.
Some more suggestions on top of deroberts' good post:
I've made it personal policy to never put fresh veggies on a pizza that are not Pre cooked. That would have to go double for thick and deep dish crusts.
If a liquid is evaporating it will hold the temperature at the evaporation point (of water in this case). This is why toast seems to get brown all at once; it takes time for the surface to heat and then evaporate away the moisture that prevents it from getting hot enough to brown. This is true of all baking where dough is involved; it is basically a drying process. So anything you can do to reduce the overall moisture content will improve the bake. This is the number one issue with pizza with veggies on it.
Deep dish modifications successful.
I ended up cooking in my 10" cast iron skillet at 475 for 20 minutes directly on my preheated pizza stone.
On top of this I followed @derobert's advice and sweat the ever loving crap out of my diced tomatoes. Was able to extract a full cup of liquid out of them AFTER draining for an hour.
The crust came out golden brown from edge to edge and all along the bottom and the exposed upper rim. There was no pool of liquid pouring through during cutting either. Was great last night and was great today for lunch.
Low moisture skim mozz! The sauce needs to be as moisture free as possible (without being solid.) You can also precook the dough without anything in it for 8-10 minutes before putting in the toppings, that will help if you have too moist sauce.
The Cooks Illustrated recipe is entirely wrong and you don't need butter layers (or any butter, for that matter, except to grease the pan). Use corn oil (3 tablespoons to every cup of AP flour); no cornmeal (there has never been cornmeal in Chicago deep dish pizza); and a very short mixing (1 minute) and kneading time (2 minutes)--no need to laminate the dough (which pizzerias do not). Proof the yeast first. It's best to par-bake the dough to set it. use a sauce made from a premium brand of ground or crushed tomatoes (6-in-1, Pagliacci, etc.) or hand-crush canned Roma tomatoes and drain thoroughly--do not cook sauce.
It seems likely one of the problems is that the cast iron is acting like a heat sink, drawing heat away from the crust instead of letting it bake. Having less moisture, as other answers mentioned, seems like it should help a bit since the heat spent to evaporate the moisture is not spent cooking - but I had a few other thoughts to offer.
You could try pre-baking the crust a bit. Just, stick it in the oven for a bit in the step between pressing the dough into the pan, and adding the toppings afterwards. It will let the crust bake on its own a little bit, and heat up the pan a little bit, so it can bake more even after the toppings are added - so you don't necessarily have to bake it until all the way done, especially if the edges might overcook that way, just enough to get the pan warmed up and the crust par-cooked (like pre-baking a pie crust).
You could try just pre-heating the pan a bit. Stick the cast iron in the oven before pressing the dough into it. You don't need it sizzling hot - that might make it hard to add the dough without scorching your fingers... but if it's heated enough to still be warm when put back into the oven, the pan can heat back up in the oven within a reasonable amount of time, it will let the crust bake, without the cast iron drawing heat away from it.
You could certainly set the pan (with or without dough) on the stove for a few minutes, to specifically heat up the bottom of the crust. This would help prevent the scenario, if you were pre-baking the crust in the oven, where the edges might overcook - since the heat is only applied to the bottom. You could use the stove to pre-heat the pan (without dough), to par-cook the bottom of the crust (with dough), or even set the whole pizza (with toppings) on top of the stove to start cooking from the bottom up, to contrast to the oven's tendency to cook top down (especially if the cast iron is cooling the bottom) - or to pre-warm the pizza to avoid thermal shock to your pizza stone, if you're putting the whole pan on top of it.
Depending on kitchen space and layout, it might even be convenient to leave the pan on the stove-top while you are layering the pizza (perhaps turned on low the whole time, or turned on at some specific point to preheat the bottom), so that pan and the ingredients are slowly warming up the whole time, and in the oven will quickly raise to temperature to start cooking .
Alternatively, you could proceed as you normally do, and just loosely tent the pizza with foil, to let it cook longer in the oven without overcooking or scorching the top of the pizza. You would want to make sure it's loose enough for moisture to escape, or it might steam your pizza toppings or leave the whole pizza too wet. This would probably be the smallest change to make in your recipe.
RE: Jolenealaska's comments--
Before you make grossly uninformed statements like this you should take the time to educate yourself. It's not at all difficult.