The most common and most likely answer to this crossword clue is the 9 letter word BEPRAISED
I've seen this crossword clue in The Guardian
B
E
P
R
A
I
S
E
D
Buddy explains!
The straight part
We know that "Lavished compliments on" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lavished compliments on = bepraised.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.