The only burek I've tasted in Chicago has come from Deta's. I like what I've had and with the exception of one out of a half-dozen times it's been baked to perfection. A recent sampling was overcooked. What I like most about Deta's is the home-like atmosphere/enviornment. Eating there is like eating in the dining room or kitchen of an old friend or family member. Soups are microwaved (from the fridge), though - and that's been a disappointment for me - though they're good tasting (but not quite "fresh"). Deta's is having a special NYE party/dinner tonight with lots of food and a 3-piece music group and if I didn't already have plans for the night I'd probably head there.
The sell them at City Fresh Market, and some of the other Balkan Markets (like Devon) and delis (like Beograd). Of the places mentioned, I would rate Deta's #1, Beograd #2 and City Fresh a distant #3. None of these places are near Wicker Park.
City Fresh Market
3201 W Devon Ave, Chicago, IL
1440 W Devon Ave, Chicago, IL
Beograd Meat Market
2937 W Irving Park Rd, Chicago, IL