The most common and most likely answer to this crossword clue is the 6 plus 6 letter word HAROLD PINTER
H
A
R
O
L
D
P
I
N
T
E
R
More clues leading to the result HAROLD PINTER
Buddy explains!

The straight part
We know that "who wrote The Birthday Party" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case who wrote the birthday party = harold pinter.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.