Heat an 8 to 10-inch non-stick omelet pan over low heat. Add 1 tablespoon of the butter and melt. While the butter is melting, break the eggs into a large bowl. Add the melted butter and whisk well. Season with salt and pepper.
Return the pan to medium-low heat. Add the olive oil and the remaining butter and heat. Add the egg mixture and grasp the pan by its handle (using a potholder if necessary to protect your hand). Start shaking the pan forward and back while stirring the eggs slowly with the back of a fork, gently lifting a moving the cooked egg so that the liquid egg slips beneath it. After about 30 seconds, the egg will have formed a uniformly cooked but still fairly moist "pancake" shape. At this time, add any filling of your choice, the tomato and goat cheese, or leave the omelet unfilled in the classic French style.
To fold the omelet, immediately tilt the pan to about a 45-degree angle by raising the handle, so that the cooked eggs nearest the handle begin to fall and fold over; you may use the fork or spatula to help this happen. Hold the far edge of the pan over a heated serving plate and continue tipping the handle up, so that the omelet folds over on itself and rolls out of the pan onto the plate. Serve immediately.