'To give the mitten to' definitions:

Definition of 'To give the mitten to'

From: GCIDE

Mitten \Mit"ten\, n. [OE. mitaine, meteyn, F. mitaine, perh. of Celtic origin; cf. Ir. miotog, Gael. miotag, Ir. & Gael. mutan a muff, a thick glove. Cf. Mitt.]
1. A covering for the hand, worn to defend it from cold or injury. It differs from a glove in not having a separate sheath for each finger. --Chaucer. [1913 Webster]
2. A cover for the wrist and forearm. [1913 Webster]
To give the mitten to, to dismiss as a lover; to reject the suit of. [Colloq.]
To handle without mittens, to treat roughly; to handle without gloves. [Colloq.] [1913 Webster]