The most common and most likely answer to this crossword clue is the 11 letter word UNDIGNIFIED
I've seen this crossword clue in The Times
U
N
D
I
G
N
I
F
I
E
D
Buddy explains!

The straight part
We know that "Embarrassing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case embarrassing = undignified.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.
What does UNDIGNIFIED mean?
UNDIGNIFIED adjective- Lacking in or damaging to dignity
- To treat without dignity.
- To demean.