Fancy pigeon that is cooking on the grill? (6)

I believe the answer is:

barbie

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'fancy pigeon that is cooking on the grill?' is the definition.
The definition suggests a singular noun which matches the answer.

This is the entire clue.

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(Other definitions for barbie that I've seen before include "Famous fashion doll" , "Popular doll" , "Outdoor party" , "Figure girl might dress up" , "Empty-headed woman" .)

I've seen this clue in The Times and The Sunday Times.

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