A young deer (noun), or to show excessive flattery and eagerness to please someone (verb).
The noun comes from Old French 'faon', from Latin 'fetus' meaning offspring. The verb meaning 'to flatter' comes from Old English 'fægnan' meaning to gladden. These likely merged because of the behavioral association with young animals acting submissive and affectionate.
It's wild that one word covers both baby deer and creepy flattery—but both involve submission and appeasing! The verb probably stuck because young animals literally 'fawn over' their parents with desperate affection to get food and protection.
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