Brother going round first restaurant (6)
I believe the answer is:
bistro
'restaurant' is the definition.
(I know that bistro is a type of restaurant)
'brother going round first' is the wordplay.
'brother' becomes 'bro' (short for brother).
'going round' means one lot of letters goes inside another (some letters go around others).
'first' becomes 'ist' (ist resembles 1st).
'bro' going around 'ist' is 'BISTRO'.
(Other definitions for bistro that I've seen before include "Small French bar and restaurant" , "Little restaurant" , "Small inexpensive eatery" , "family concern perhaps" , "Small bar-cum-restaurant" .)