is lambskin with the wool dressed outwards, worn on the edge of capes, bachelors' hoods, and so on. Budge Row, Cannon Street, is so-called because it was chiefly occupied by budge-makers.
O foolishness of men! that lend their ears To those budge-doctors of the stoic fur.
Milton: Comus, 706, 767.
(To) is the French bouger, to stir.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894