Cut the unsalted butter into cubes and melt gently in a small saucepan over low heat until fully melted. Skim off foam if desired and keep warm (do not let brown).

Prepare double-boiler

Set a heatproof bowl over a saucepan of barely simmering water (water should not touch the bowl). Keep the water at a gentle simmer—too hot will scramble the yolks.

Whisk yolks

In the heatproof bowl, whisk the 3 egg yolks with the lemon juice, Dijon (if using), and a pinch of salt until slightly thickened and pale, about 1 minute.

Cook gently

Place the bowl over the simmering water and whisk continuously until the yolks are warm and thick enough to coat the back of a spoon, about 2–3 minutes. Do not let them get too hot.

Emulsify butter

Remove the bowl from heat. Very slowly drizzle the warm melted butter into the yolks while whisking constantly to create a smooth emulsion. Begin drop by drop, then increase to a thin steady stream as the sauce thickens.

Finish and season

Once all butter is incorporated, whisk in lemon juice, cayenne, and 1/4 tsp salt. If the sauce is too thick, whisk in warm water 1 tsp at a time until desired consistency is reached. Keep the hollandaise warm by placing the bowl in a warm spot or over the warm (off) pan; do not reheat over direct heat.

For the eggs & assembly

Toast English muffins

Split the English muffins and toast or grill until golden. Spread 1/2 tablespoon unsalted butter on each toasted half while hot. Keep warm.

Poach eggs

Fill a wide shallow saucepan with 6–8 cm (2–3 in) of water and bring to a gentle simmer. Add the white vinegar. Create a gentle whirlpool with a spoon. Crack one egg into a small bowl or cup, then gently slide it into the center of the whirlpool. Repeat with remaining eggs, poaching at most two at a time so they have space. Poach eggs until whites are set but yolks remain soft, about 3–3½ minutes for large eggs. Use a slotted spoon to lift eggs, gently blot excess water on paper towel.

Season eggs

Lightly season the poached eggs with a pinch of salt and a grind of black pepper.

Assemble

Place two toasted English muffin halves on each plate. Top each muffin half with about 30 g smoked salmon (adjust to taste). Place a poached egg on top of the salmon. Spoon 2–3 tablespoons of warm hollandaise over each egg, allowing it to run down the sides. Sprinkle minced chives and an extra pinch of black pepper over the top. Serve immediately.