I don't know of any Mexican restaurants, taquerias, or burrito joints in Chicago that CAN'T serve your burrito in the manner you've described. As I recall, most places refer to this style as "suiza"; with cheese and, as BT states, topped with red or green sauce. Although around here, suiza usually indicates red sauce. It is available not only on burritos but enchiladas and flautas as well. If you don't see "suiza" offered on the menu, ask for it
There are several burrito threads here at Roadfood, some of them dealing specifically with Chicago's many burrito locations. You should use the search function and dig them up; they contain some good information.
As for my favorites, if I want a sit down experience, I'll go to Tecalitlan (TECKa-leedle-LAN) on Chicago Avenue about 5 blocks west of Ashland. When I worked in the city, my buddy and I would go there for lunch once a week. I'd order the steak burrito, he'd order the avocado. Then we'd split them and each take half, thereby fooling ourselves that we were getting a balanced meat and veggie lunch(HAH). Didn't matter, they were great. An added plus at Tecalitlan is their pico de gallo that is served at every table as soon as you sit down. The heat level would occasionally vary depending on who made that day's batch. But flavor-wise, it was always brilliant!
These days when I want my burrito to go, I will stop in at any one of the numerous "Burrito House" locations around the city. They make an outstanding version in both the typical "big as your head" size called the "king" and, thankfully, in my advancing years, a more manageable "junior" size, which frankly ain't that much smaller than the "king".
Although burritos are more likely to be found in an American Mexican restaurant, they frequently will reflect the original regional Mexican preferences of the store owner serving them. Thus, at Burrito House, if you want cilantro or onions, you have to ask for them. At Tecalitlan they are automatic. Both places charge extra for avocado on a burrito (unless of course it's an avocado burrito).
Not sure what else to tell you except, welcome to Roadfood. You seem to be at an interesting point in your life; your stated love of both Chi-Chi's and Chipotle indicate a need for a "safety zone" in your dining experience. On the other hand, the fact that you've come to Roadfood indicates that maybe your ready for a little adventure; take some chances. The nice thing about the recommendations and the opinions expressed here, is that they allow you to do both; have a food adventure, still knowing that someone whose opinion you trust has already been there and has given it a thumbs up. Look around there's lot's of good information and converstion to be had here.