The most common and most likely answer to this crossword clue is the 3 plus 2 plus 3 plus 7 letter word TOP OF THE MORNING
I've seen this crossword clue in The Daily Telegraph
T
O
P
O
F
T
H
E
M
O
R
N
I
N
G
More clues leading to the result TOP OF THE MORNING
Buddy explains!
The straight part
We know that "11.59 am greeting?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case 11.59 am greeting? = top of the morning.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.