Green’s Dictionary of Slang

(US) snobbily, pompously; in a boastful manner.

1869		Sportsman (London) 3 June 2/1: Notes on News [...] The water-leap been prepared [...]’ said, in the most solemn manner, an evening paper regarding the [...] show at the Agricultural Hall. And yet the same journal does very high falutin when writing about the water-jump at a steeple-chase.
1902		C.L. Cullen More Ex-Tank Tales 136: Ef thet feller kin write as highfalutin’ as he kin talk, he’s th’ man.
1906		A. Adams ‘In the Hands of His Friends’ in Cattle Brands 🌐 Jack must have pictured this claim mighty hifalutin to that gal, for she’s a way up good-looker.