Down by Law. Jim Jarmusch, 1986.
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Edition screened: Criterion Blu-ray #166, released 2012. English language. Runtime approximately 107 minutes.
Summary: Rabbit hunting and cooking.
Details:
1) Roberto Benigni has caught a rabbit in the woods, and holds the small dead rabbit by the ears, 1:21:24-1:21:55.
2) We see a rabbit being roasted on a spit during dialogue, 1:23:05-1:26:31.