Start by melting the butter in a small saucepan over low heat—this gentle heat prevents it from browning and keeps it at the right temperature for emulsifying. While the butter melts, whisk together the egg yolks, fresh lemon juice, and mustard in a medium heatproof bowl until the mixture becomes pale and slightly thickened, about 1-2 minutes. I like to use fresh lemon juice rather than bottled because the acidity is more vibrant and helps create a silkier sauce.

Set the bowl with the egg yolk mixture over a pot of gently simmering water (the bowl should not touch the water), creating a double boiler. Whisk constantly while slowly drizzling in the melted butter—this gradual incorporation is crucial for emulsifying the sauce and creating that creamy, silky texture. Continue whisking for 8-10 minutes until the sauce is thick and pale. Once all the butter is incorporated, remove from heat and season with salt, cayenne pepper, and a pinch of black pepper. If the sauce breaks or becomes too thick, whisk in a tablespoon of warm water at a time until it reaches the right consistency.

While the sauce finishes, slice the croissants horizontally and set them on a baking sheet. Turn on the broiler to preheat. Toast the croissant halves under the broiler until they're golden and crispy, about 2-3 minutes—watch them closely as they brown quickly. Remove from the oven and set aside. While the croissants toast, you can arrange the ham slices nearby for quick assembly.

Fill a large, shallow pan with about 3 inches of water and bring it to a gentle simmer over medium heat. Add the vinegar to the water—this helps the egg whites congeal faster and keeps them from spreading. Once simmering, carefully crack each egg into the water one at a time, spacing them apart so they don't touch. Poach for 2.5 to 3.5 minutes, depending on how runny you like the yolk. I prefer closer to 2.5 minutes to keep the yolk nice and jammy, but adjust based on your preference. Use a slotted spoon to transfer the poached eggs to paper towels to drain any excess water.

Place a toasted croissant bottom on each serving plate. Layer 2 slices of ham and then 1 poached egg on top of each croissant bottom. Spoon the warm hollandaise sauce generously over the eggs. Top with the toasted croissant top and garnish with finely snipped chives and a crack of fresh black pepper. Serve immediately while everything is still warm—the contrast between the warm croissant, creamy sauce, and runny yolk is what makes this dish special.