8-inch seasoned omelet skillet (like black steel or cast iron)
Preheat the broiler and arrange the oven rack in the highest position.
Put the eggs, half-and-half, salt and pepper in a bowl and whisk until frothy.
Melt the butter in the skillet over medium-high heat. Pour in the egg mixture and let sit until the bottom starts to cook. Using a rubber spatula, slowly push the cooked egg from 1 side of the skillet to the other, allowing the raw egg to reach the bottom and cook--this creates fluffiness and keeps the bottom from burning.
When the omelet is cooked on the bottom and the top is still runny, place the skillet just under the broiler with the oven door open and the handle sticking out, and broil until the omelet rises and browns lightly, about 1 minute. (Alternatively, flip the omelet in the skillet and cook briefly, to finish.)
Scatter the cream cheese and smoked salmon over the omelet; then sprinkle with the onion and capers. Broil again to warm the topping, about 30 seconds
Transfer the omelet to a cutting board, and cut into 4 triangles. Divide the omelets among plates and garnish with the lemon slices and dill.