Smoked beef over rye, initially given to French friend (8)
I believe the answer is:
pastrami
'smoked beef' is the definition.
(type of smoked meat)
'over rye initially given to french friend' is the wordplay.
'over' becomes 'past' (both can mean finished).
'initially' suggests taking the first letters.
'given to' says to put letters next to each other.
'french friend' becomes 'ami' (I've seen this before).
The initial letter of 'rye' is 'r'.
'past'+'r'+'ami'='PASTRAMI'
(Other definitions for pastrami that I've seen before include "Highly seasoned cut of smoked beef" , "Seasoned smoked beef, served in thin slices" , "Thinly sliced beef" , "Spiced sliced beef" , "Spicy smoked beef" .)