The most common and most likely answer to this crossword clue is the 8 plus 5 letter word ELECTRIC CHAIR
I've seen this crossword clue in The Times
E
L
E
C
T
R
I
C
C
H
A
I
R
Buddy explains!
The straight part
We know that "method of dispatch" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case method of dispatch = electric chair.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.