This technique is based on a classic French omelet, which is moist and fluffy without even a hint of browning. Note that it’s cooked in three stages, at three different heat settings.
Crack eggs into a bowl; add salt and pepper. Whisk until the mixture is very loose and very little egg clings to the whisk when you lift it out, about 1 minute.
Roll basil leaves tightly into a ball and make thin slices that produce small ribbons of basil. Cut tomatoes into ¼-inch cubes. Set basil and tomatoes aside.
Heat butter in a nonstick fry pan over moderately high heat. When butter begins to sizzle but before it melts completely, swirl it around the pan and pour in eggs. Immediately turn the burner to medium. Use a wooden spoon to gently and continuously scrape the sides and bottom of the pan. Meanwhile shake the pan to keep the egg mixture flat, like a pancake. As the egg pancake cooks, the edges will become very ragged. At this point, use the spoon to push some of the uncooked egg from the middle toward the edge to roughly fill it in, finishing the egg pancake. Take it off the heat but leave burner on.
Next, place goat cheese and other fillings in a line down the middle of the omelet; return to heat. Turn heat to low. Use a spatula to loosen one edge of the omelet and flip it over, covering the filling. Leave undisturbed 30 to 45 seconds. Serve on a warmed plate.