Ethan Hawke delivers the performance of his career in Robert Budreau’s melancholy but fascinating biopic about jazz trumpeter Chet Baker. Sprung from an Italian prison in the 1960s by a Hollywood filmmaker, Baker stars in a movie-within-the-movie that’s based on his own past as a womanizer and heroin addict, and in the process he charms the actress playing his ex-wife (Carmen Ejogo of Selma). But when Baker loses his front teeth in a brawl with drug dealers, the injury destroys his embouchure and threatens to ruin his career. Hawke channels Baker’s real-life eccentricities, and Ejogo brings warmth and levity to her role. Long takes, naturalistic dialogue, and a loose narrative structure make for a film that feels as improvised and authentic as its music.