Step 1: Clarify the butter. In a small saucepan over low heat, gently melt the unsalted butter. Skim off any foam that rises to the top. Continue to heat gently until the butter separates into clear golden liquid and white milk solids at the bottom. Carefully pour the clear butter into a separate measuring cup, leaving the milk solids behind. Keep the clarified butter warm.
Step 2: Prepare a double boiler. Fill a medium saucepan with about 1-2 inches of water and bring it to a gentle simmer over medium-low heat. Ensure the heat-proof bowl you'll use sits snugly on top without touching the water.
Step 3: Whisk the yolks and lemon juice. In the heat-proof bowl, combine the egg yolks, fresh lemon juice, and a pinch of salt. Place the bowl over the simmering water.
Step 4: Cook the yolks (sabayon). Whisk the egg yolk mixture continuously and vigorously. The mixture will start to thicken, become pale, and increase in volume. This process should take about 3-5 minutes. Ensure the eggs don't scramble by whisking constantly and lifting the bowl off the heat for short intervals if it gets too hot. The mixture should be thick enough to leave a trail when you lift the whisk.
Step 5: Emulsify with butter. Once the egg mixture is thick and pale, remove the bowl from the heat (but keep it over the empty saucepan to retain warmth). Slowly, drop by drop at first, begin to whisk in the warm clarified butter. Continue whisking constantly. As the sauce starts to emulsify, you can add the butter in a thin, steady stream, still whisking continuously, until all is incorporated.
Step 6: Season and serve. The hollandaise sauce should be thick, smooth, and glossy. Stir in the optional pinch of cayenne pepper and taste for seasoning, adding more salt or lemon juice if needed. Serve immediately.