This is the recipe I use when I teach this sauce:
"Bearnaise sauce
Essence:
Chop finely:
1 shallot
2 tablespoons fresh tarragon
1 tablespoon fresh parsley
Put in a small sauce pan with
¼ cup tarragon vinegar
¼ cup water
Boil down to a couple of tablespoons of liquid, and allow to cool. Do not put essence in the sauce hot, or it will separate and ruin the sauce.
The tarragon and vinegar are the key elements here, but proportions vary widely from cook to cook. Some use white wine instead of water, some use much more tarragon and no parsley, some use spring onions instead of shallots, etc. It doesn’t really matter as long as the tarragon and vinegar are there.
Make Hollandaise base sauce:
Everyone uses a double boiler, which is a good idea. Ignoring caution, put four egg yolks into a sauce pan and put on low heat, stirring constantly and being certain to scrape the bottom of the pan so no part of the sauce stays in contact with the pan for too long. A “spoonula” works best.
As soon as the eggs are on the heat, start adding 2 sticks of unsalted room-temperature butter a tablespoon or so at a time, stirring until the sauce thickens noticeably -- about 20 minutes. Add essence to taste and re-thicken. Remove from heat and keep stirring until no longer hot, or the sauce will separate. Serve onto or beside meat that is not too hot, or, again, the sauce will separate. Some sauce on the side is good, too.
This process cannot be learned from print. It takes experience. Everyone has sauces separate, in which case you have scrambled eggs floating in butter – horrible -- throw it out and start over. It is unwise to start a Hollandaise or Béarnaise without enough eggs and butter for a do-over.
There are other ways to make this sauce, such as using a blender or food processor or whisk to emulsify, but those sauces are not as good. Thickening by cooking is preferred."