The most common and most likely answer to this crossword clue is the 16 letter word WELL-UPHOLSTERED
I've seen this crossword clue in The Times
W
E
L
L
-
U
P
H
O
L
S
T
E
R
E
D
Buddy explains!
The straight part
We know that "fat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fat = well-upholstered.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does WELL-UPHOLSTERED mean?
WELL-UPHOLSTERED adjective- Fat.