The "not Walter's" relates to the fact that Chambers gives Walter Scott's spelling of the word as 'brouze', which would otherwise have been a perfectly good alternative solution to the clue.
It seems that a broose is (or was) "a race on horseback, or on foot, by the young men present at country weddings in the north, the course being from the place where the marriage ceremony is performed (in Scotland the bride's former home) to the bridegroom's house."