If I could bend the rules, I would propose the following Chicago themed double feature:

"Man with the Golden Arm" (1955)

Adaptation of the Nelson Algren novel. Directed by Otto Preminger. Stars Frank Sinatra and Kim Novak.

Chicago connections: Algren, Novak, and an uncredited screenwriter Ben Hecht.
Chicago based author Algren detested the film and the director. Novak was a Chicago native. Hecht was a former Chicago newspaperman. Great musical score.

"Compulsion" (1959)

Directed by Richard Fleischer based upon the Meyer Levin novel (a thinly disguised retelling of the 1924 Chicago kidnapping and murder of Bobby Franks by a pair of intellectual thrill killers Nathan Leopold and Richard Loeb). Stars Orson Welles, Dean Stockwell, Bradford Dillman and E. G. Marshall.