Doctor keen on eating last of black toast (5,2)
I believe the answer is:
drink to
'toast' is the definition.
('drink to' can be a synonym of 'toast')
'doctor keen on eating last of black' is the wordplay.
'doctor' becomes 'dr' ('Dr' can be a synonym of 'doctor').
'keen on' becomes 'into' (I've seen this before).
'eating' means one lot of letters goes inside another.
'last of' suggests the final letters.
The last letter of 'black' is 'k'.
'into' enclosing 'k' is 'inkto'.
'dr'+'inkto'='DRINK TO'
(Another definition for drink to that I've seen is " Toast".)