For deep dish - Taste of Chicago in Addison was the closest I could find to pizza I'd had in The Chi. No it's not as good as Lou Malnati's but it is definitely better than most spots that make deep dish.
For NY-style, there's quite a bit more variety. Personally, I think for just pizza...Piggie Pies on Greenville in Dallas is the best. Lots of cheese, quality toppings and friendly staff. It's my "go to" spot. Only issue with them is I feel their prices in have gone up but the quality of their non-pizza offerings has dipped a bit.
For the super, cracker thin crust that people in Dallas like...Fireside Pies probably has some of the best food. It's tastes better at the restaurant though. If you take it home or eat it the next day, not quite as good. The triple-roni is amazing. Love the bruschetta and meatballs.
iFratellis is also a winner. Great pizza for the cracker-thin crowd.