The most common and most likely answer to this crossword clue is the 7 letter word ROSIEST
I've seen this crossword clue in The Daily Telegraph
R
O
S
I
E
S
T
Buddy explains!
The straight part
We know that "most embarrassed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most embarrassed = rosiest.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 ROSIEST mean?
ROSIEST adjective- Rose-coloured.
- Resembling rose, as in scent of perfume.
- Optimistic.