1. Place the milk and onion in a small saucepan. Bring to a simmer over medium low heat; simmer for 5 minutes until onion is soft. Drain milk and reserve onion.
2. Preheat the broiler. Whisk 3 of the eggs together with some salt and pepper. Heat an 8” to 9” nonstick frying pan over medium high heat, then add 1 Tbsp. butter. Swirl the butter around the pan until foaming subsides. Pour in the eggs. As the eggs begin to set, use the tines of a fork to “catch” the edges of the omelet, pulling up folds of the egg and letting uncooked egg run underneath.
3. When the omelet is set on the bottom but still very wet on top, spread half the onion mixture over the eggs. Immediately lay half the smoked salmon pieces over the eggs. Drizzle with 2 Tbsp. heavy cream, dust with 2 Tbsp. Parmesan cheese, and place the pan under the hot broiler.
4. Broil until golden brown and puffy. Meanwhile, whisk the remaining 3 eggs with salt and pepper. When golden brown, slide the finished omelet onto a warm plate, folding it into a half round.
5. Return pan to burner, add remaining butter, and cook second omelet as above.
6. Serve with slices of rustic bread and fresh fruit.