The most common and most likely answer to this crossword clue is the 4 plus 7 letter word LOVE HANDLES
I've seen this crossword clue in The Guardian
L
O
V
E
H
A
N
D
L
E
S
More clues leading to the result LOVE HANDLES
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 = love handles.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.