Verb-object construction:
打 ('to make/perform') +
手势 ('hand-sign'). The two parts can split —
打了个手势 ('made a gesture'),
打什么手势 ('what gesture did you make'). Compare
打电话 ('to make a phone call'),
打喷嚏 ('to sneeze'),
打招呼 ('to greet') — all V-O patterns where
打 carries the lightverb 'to perform' meaning.
