Cooked over direct heat using a grill; or intensely questioned about something.
From Old French 'greil' (grating/grid). The cooking method meaning came first (food on a grated surface over fire). The 'questioning' sense emerged in the 1900s from the idea of cooking someone 'under heat.'
The jump from cooking to interrogation happened because of a perfect metaphor: cops putting suspects 'under the heat' of questioning borrowed the sizzle and pressure of grilling food—language loves to cook up new meanings!
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