Prep everything first (mise en place). Crack the 6 eggs into a medium bowl, add 2 tablespoons whole milk and 3/4 teaspoon of the salt, and beat gently until just combined. Beating incorporates a little air for silkiness but avoid over-whipping — you want a uniform pale mixture, not a froth, to keep the omelette tender.

Make the garlic breadcrumbs: tear or pulse the 2 slices of stale bread into coarse crumbs. Heat a small skillet over medium heat and add 1/2 tablespoon olive oil. Add the breadcrumbs and toast, stirring frequently, until golden and crisp (about 4–5 minutes). Add the minced garlic in the last 30 seconds so it softens and becomes fragrant without burning. Transfer to a bowl and season with a pinch of salt. Toasting the crumbs ahead gives a crunchy textural contrast to the soft eggs.

Caramelize the onion: in the same skillet, add the remaining 1/2 tablespoon olive oil and the thinly sliced onion with a pinch of salt. Cook over medium-low heat, stirring occasionally, for 8–10 minutes until the onion is deeply golden and sweet. Lower heat and be patient — slow cooking converts starches to sugars, producing the sweet, rich flavor that contrasts the eggs.

Brown the butter for finishing: in a small saucepan or another small skillet, melt 2 tablespoons of the butter over medium heat. Swirl frequently; after a few minutes the milk solids will foam and then turn golden-brown with a nutty aroma. Immediately remove from heat to prevent burning and set aside. Browned butter adds toasty, savory complexity that dresses the finished omelette.

Cook the omelette base: heat a 10-inch nonstick skillet over medium-low. Add the remaining 1 tablespoon butter and let it melt, coating the pan. When the butter is foamy but not browned, pour in the beaten eggs. Using a silicone spatula, stir gently in small figures-of-eight to form soft curds for the first 20–30 seconds, then stop stirring and let the eggs sit for about 10 seconds to set. Low heat and constant attention produces a custardy, tender curd rather than a dry, rubbery omelette.

Shape and fold: when the eggs are mostly set but still slightly glossy on top (about 30–45 seconds more), spread the caramelized onions across the center and sprinkle half the garlic breadcrumbs over them. Tilt the pan away from you and use the spatula to fold one edge over toward the center, then fold the other edge to form a classic French tri-fold or simple half-moon, depending on your comfort. The goal is to keep a soft interior while creating a neat exterior.

Finish with browned butter: slide the omelette onto a warmed plate seam-side down. Spoon the reserved browned butter over the top — this adds aroma and a glossy finish. Scatter the remaining garlic breadcrumbs and chopped parsley on top for crunch and a fresh note. Resting briefly (30–60 seconds) lets residual heat finish cooking the interior without overcooking.

Season and serve: taste for salt and pepper, adjusting as needed. Serve immediately while the omelette is warm, with extra toasted breadcrumbs on the side if you like an even crispier bite. Immediate serving preserves the contrast between the warm, silky egg interior and the crisp breadcrumb topping.