Melt the butter, juice one lemon, and separate the egg yolks from the whites. Put the yolks into a stainless steel bowl or small saucepan; reserve the whites for another use.
Pour 1 tablespoon of fresh lemon juice into the egg yolks.
Place the bowl or pan over a gentle heat by holding the bowl with an oven mitt over the stove burner set at low to medium low heat. Constantly whisk the yolks while allowing the heat to rise slowly. Lift the whisk high in the bowl to whip air into the eggs. Also, lift the bowl occasionally to give the eggs a break from the heat while constantly whisking. You want to be careful not to scramble the eggs.
You will notice that the mixture starts to resist the whisk slightly more and it will be thick enough that it coats the bottom of the bowl when it’s tilted. Also when you lift the whisk out of the mixture some of it will stick to the whisk and fall back into the bowl in ribbons.
The eggs will cook and thicken in one to three minutes. The mixture should be creamy.
Once the yolks are thickened and you can see small wisps of steam rising out of them take the bowl off of the heat immediately and continue to whisk quickly for at least another 30 seconds in order to keep the yolks from cooking too much. Put the bowl down on a pot holder or wooden cutting board because it will still be hot.
Undercooking the sauce results in a mixture that’s too thin; overcooking it creates lumps. If it curdles, start over with new egg yolks.
Now it is time to emulsify to butter into the egg yolk mixture. Whatever you do, do NOT dump all that butter into your egg mixture.
Add the butter drop by drop at first. As the emulsion forms, the butter can be added in larger amounts. Now add about a teaspoon amount of melted butter and whisk it in quickly; repeat five more times. Increase the butter to one tablespoon at a time, whisking to incorporate thoroughly until all of the butter is used up.
The texture should be smooth and creamy with no hint of oiliness and no lumps. It should be very thick yet pourable. If necessary, thin the sauce with a few drops of warm water.
Now all that remains is to season it. Add a dash of cayenne pepper or white pepper and a pinch of salt to taste; mix into the sauce well.
Serve immediately; do not heat or chill the sauce.