Green’s Dictionary of Slang

(US) arrogantly, aggressively.

1899		Ade Fables in Sl. (1902) 90: All during the Seventeen Years Zoroaster and Zendavesta continued to walk chesty and tell People how Good they were.
1946		Mezzrow & Wolfe Really the Blues 61: [We] walked around so chesty we would have made Miss Peacock pull a fade-out.