Crack the eggs into a bowl. Do not whisk yet. The goal is to mix them in the pan, not beforehand, to achieve a marbled texture.

Place a non-stick pan over medium-low heat. Add the cold butter cubes. Once the butter is melted and foamy, pour in the eggs.

Immediately begin stirring the eggs constantly with a rubber spatula, scraping the bottom and sides of the pan. The key is to cook the eggs very slowly and gently, preventing the formation of large curds.

Continue stirring and scraping for about 5-7 minutes, or until the eggs are mostly set but still very wet and creamy. They should look like a thick, pale yellow sauce.

Remove the pan from the heat. Stir in the crème fraîche, salt, and pepper. The residual heat will finish cooking the eggs to the perfect, custardy consistency.

Spoon the creamy eggs onto the toasted sourdough. Garnish generously with fresh chives and a pinch of Espelette pepper, if using. Serve immediately.