The most common and most likely answer to this crossword clue is the 6 letter word CROWED
I've seen this crossword clue in The Guardian
C
R
O
W
E
D
More clues leading to the result CROWED
Buddy explains!

The straight part
We know that "Boasted" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case boasted = crowed.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 CROWED mean?
CROWED verb- To make the shrill sound characteristic of a rooster; to make a sound in this manner, either in gaiety, joy, pleasure, or defiance.
- To shout in exultation or defiance; to brag.
- To test the reed of a double reed instrument by placing the reed alone in the mouth and blowing it.