Cheese burger's closing the branch (5)
I believe the answer is:
brier
'branch' is the definition.
(brier is a kind of branch)
'cheese burger's closing' is the wordplay.
'cheese' becomes 'brie' (Brie is an example).
'closing' suggests the final letters (the letter which ends or closes the word).
The last letter of 'burger' is 'r'.
'brie'+'r'='BRIER'
'the' acts as a link.
(Other definitions for brier that I've seen before include "Thorny bush like the wild rose" , "Prickly shrub" , "plant" , "It may be smoked" , "smoking pipe" .)