The most common and most likely answer to this crossword clue is the 5 plus 6 letter word MARIA CALLAS
I've seen this crossword clue in The Times
M
A
R
I
A
C
A
L
L
A
S
Buddy explains!
The straight part
We know that "I strike up, moving La Scala" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case i strike up, moving la scala = maria callas.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.