Why an emulsion breaks

Fat suspended in water, held there by the yolk. Three things break it: too much heat, too much fat at once, too little liquid.

1. Heat. Above about 65 °C the yolk proteins set and stop holding anything. Off the heat, bowl warm to the hand, never hot.
2. Speed. The first third of the butter has to go in almost drop by drop. Once the emulsion is established it will take the rest in a thin stream.
3. Water. A tablespoon of it is what the fat hangs on to. A sauce that has gone thick and dull is usually short of water, not short of whisking.
4. The rescue: a clean bowl, a tablespoon of hot water, and the broken sauce whisked into it a spoonful at a time. You are not fixing the old emulsion, you are starting a new one.