Seems here is a definition of the term:
http://dare.wisc.edu/words/WotM/cut-pigeon-wing

cut the pigeon('s) wing
cut the pigeon(’s) wing v phr Also cut a pigeon wing [cut to dance + pigeon wing a dance movement imitative of pigeons] esp S Midl
To execute intricate dance steps gracefully.
1897 (1952) McGill Narrative 139, In many of these dances an opportunity is given for the display of much grace and artistic coquetry by the young ladies, and of activity by some of the young men, as they “cut the pigeon wing.”1898 Lloyd Country Life 15 AL, I use to cut the pigeon wing around Miss Tildy some myself. 1912 DN 3.574 wIN, Cut a pigeon wing. . . To dance with graceful steps. 1927 AmSp 2.352 WV, Cut the pigeon’s wing. 1940 FWP Guide TX 114 (as of 1828), When young people danced in those days, . . they ‘shuffled’ and ‘double-shuffled,’ and ‘wired’ and ‘cut the pigeon’s wing,’ making the splinters fly. 1946 Greer–Petrie Angeline Steppin’ 35 csKY, He’d jump out in the middle of the floor and cut the pidgeon wing. 1954 Harder Coll. cwTN, Cut the pigeon wing. . . To dance in a fancy way. “ ’At air girl shore is a-cuttin’ the pigeon wing.” 1966 DARE (Qu. FF5a, . . Steps and figures in dancing—in past years) Inf DC8, Cut the pigeon’s wing.