What is an egg emulsion?
To know what's happening here you should know what "emulsifying" is.

First we need to know that fat is "non-polar" substance. what does that mean?

It means that all fats can't be dissolved in water which is "polar" substance.

So how can we make fat -which is non-polar- dissolve in water or any other polar substances?

We add "emulsifiers" like: egg yolk and many other chemical substances like Lecithin.

These emulsifiers are (polar and non-polar) so they can link fat with water together.

We add butter slowly so that the molecules of emulsifier can link the molecules of butter with molecules of sauce -which is polar-.

If there's no free molecules of emulsifier, every molecule of added butter will not be dissolved in the sauce.