I made butter poached lobster tail today. Every recipe on the internet warns not the boil the butter when making the beurre monté, otherwise the butter will separate. No recipe explained why this is such a bad thing. If you're just cooking the lobster in the butter, why does it matter if it separates or not?
Cooking doesn't equal boiling, mind. For example, sous-vide cooking cooks something at very low temperature.
Beurre monté is a sauce of emulsified butter, which will remain emulsified up to 80-85ºC (more or less). If the temperature goes up, your sauce will separate into fat (~80%) and water (a little less than 20%), will lose texture and you also might end up with rancid flavors if the temperature goes up too much and the fatty acids in butter hydrolize (as in break down with the help of water)