E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To hang out the besom. To have a fling when your wife is gone on a visit. To be a quasi bachelor once more. Taking this in connection with the following phrase, it evidently means, holding the marriage service in abeyance.
This is French argot. Rotir le balaí (to burn the besom) means to live the life of a libertine, whence balochard, Paris slang for a libertine. Probably our phrase, burn the bellows, is pretty much the same as rotir le balai.
Jumping the besom. Omitting the marriage service after the publication of banns, and living together as man
and wife. In Southern Scotch, a street-walker is called a besom, and in French balai (a besom) means the life of a libertine, as Rôtir le balai; Il ont bien rôti le balai ensemble, where balai means a debauch or something worse. No further explanation can be needed or could be given.