Browned off

[hughcompcan2]

Where does the expression originate

[Quizmonster]

According to Partridge's 'Dictionary of Slang' it was regular army slang from the time of World War 1 and then 'taken over' by the RAF in the late 1920s. However, there is no actual written evidence of it prior to the late 1930s. There is no firm evidence of precisely why it came to mean 'bored/fed up'.
Purely as a guess, how about the idea of meat in the final stages of cooking being 'browned off'' before being put on a plate and eaten? The idea might just be a reference to something reaching "the end of its tether", as it were. But that is just pure guesswork.

[searchme]

Brewers cites 'Cheesed off' as a similar expression widely current in World War 2 and one of the many expressions associated with cooking.

[danube]

'Cheesed off' - possibly derived from Charles Dickens' The Old Curiosity Shop where Dick Swiveller is asked, 'How is the cream of the clerkship?' He replies, 'Turning rather sour, I'm afraid, almost to cheeseness'

Bored, discontented