I must send you to Birchin Lane, i.e. whip you. The play is on birch (a rod).
“Passing through Birchin Lane amidst a camp-royal of hose and doublets, I took ... occasion to slip into a captain's suit- a valiant buff doublet stuffed with points and a pair of velvet slops scored thick with lace.” —Middleton: Black Book (1604).
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894