Green’s Dictionary of Slang

eyebrow n.

In phrases

walk off on (one’s) eyebrow (also waltz off on (one’s) eyebrow)

to go away; often as imper.

1874		Manchester Eve. News 14 Sept. 4/3: If he don’t stop trying to play roots on me, I’ll put a Mansard roof on him and make him waltz off on his eyebrow.
1876		St Louis Globe-Democrat 19 Jan. n.p.: The bar-keeper gets ‘riled’ [...] and then adds ‘take a walk,’ ‘skirmish,’ ‘mosey,’ ‘walk off on your eyebrow’.