Cooking food over direct heat, or questioning someone intensively and relentlessly.
From French griller 'to broil', from Old French greille 'grating, grill'. The interrogation sense developed from the metaphor of subjecting someone to intense 'heat' through persistent questioning, similar to how food is subjected to direct flame.
The connection between cooking and interrogation in 'grilling' reveals how we use physical experiences to understand social ones - both involve applying sustained pressure or intensity to achieve a desired result. It's wonderfully apt that we use the same word for both extracting flavors and extracting information!
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