The most common and most likely answer to this crossword clue is the 9 letter word PAN-FRIED
I've seen this crossword clue in The Times
P
A
N
-
F
R
I
E
D
Buddy explains!
The straight part
We know that "cooked in shallow fat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cooked in shallow fat = pan-fried.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 PAN-FRIED mean?
PAN-FRIED verb- To fry in a pan with minimal oil.