Seeking to get astonishingly fat on a half of beer! (5)
I believe the answer is:
after
'seeking' is the definition.
(I've seen this in another clue)
'astonishingly fat on a half of beer' is the wordplay.
'astonishingly' is an anagram indicator.
'on' is a charade indicator (letters next to each other).
'a half of' means to take half the letters.
'beer' with half the letters taken is 'er'.
'fat' with letters rearranged gives 'aft'.
'aft'+'er'='AFTER'
'to get' is the link.
(Other definitions for after that I've seen before include "In the time following" , "Subsequent to" , "since" , "Post, later than" , "Following, consequent to" .)