I live in nearby Indiana, so I am hoping that there might be somewhere in Chicago (or the USA!) that makes Döner Kebabs in the way that Turks in Germany do.
To be clear: Döner Kebabs are sandwiches somewhat like the much more available Greek-style shawarmas or gyros. Lamb and beef are thinly sliced, layered onto a vertical rotisserie, and cooked from the side. Then seared meat is shaved off to order from the stack. In Berlin, where I liked them best in Germany, the meat was shoveled into a bread that was more bread-like than pita (Fladenbrot), and it was garnished with lettuce, sliced cabbage, onions, cukes, and maybe tomato, and we would typically use a spicy red sauce (along with, or instead of the mild and white Greek-style tsatsiki sauce which we typically se in the USA). I attach a picture of the real German thing.