The most common and most likely answer to this crossword clue is the 4 plus 4 letter word OVER EASY
I've seen this crossword clue in Financial Times
O
V
E
R
E
A
S
Y
Buddy explains!
The straight part
We know that "How eggs may be served" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case how eggs may be served = over easy.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.