Why it happens
Beurre blanc is an emulsion of butter in a reduced mixture of wine, vinegar and shallots. Unlike hollandaise, it has no egg yolk to hold it together, so it relies on the milk solids and water in the butter to keep the fat suspended. If the sauce gets too hot, the butter melts completely and the fat separates out, so the sauce breaks. If it gets too cold, the butter firms up and the sauce turns thick and lumpy instead. Adding the butter too fast can also overwhelm the reduction.