Broue is used only in French Québec (the proper word being mousse) and in no other French speaking country. I can only find it in my dictionary of Québec colloquialisms. The etymology is not included.

I always assumed it was extrapolated from brouiller, which means mix up vigorously. As with most things that are mixed up vigorously, (egg whites for example), a foam is created.

Brew, as a term for beer, is not commonly used by Québec Anglophones. I wonder if that term is a U.S. thing. And, one thing you do not what on your brew is a lot of broue. Fire THAT bartender