Separate the egg whites from the egg yolks. Beat the egg whites with a pinch of salt until firm peaks form (this means that if you turn the whisk upside-down, the foam will keep its shape and stay firmly attached to the whisk). Make sure to not overbeat the egg whites.
In a small bowl, beat lightly the egg yolks with sugar and vanilla extract. Add them slowly to the whipped egg whites, mixing at the lowest speed until incorporated.
Add the flour, cocoa, and finely chopped chocolate (it can also be grated). Fold in gently with a silicone spatula.
Heat the butter in a non-stick pan (8-inch/21-22 cm). When it's sizzling, pour in the omelet batter. Spread with a spatula in an even layer.
Cover the pan with a lid (or a piece of aluminum foil or big chopping board) and cook over lowest heat for about 3-5 minutes until the omelet is golden at the bottom (take a peek by lifting it a bit with a silicone spatula).
Using two wide spatulas, turn the omelet to the other side: first separate the edges of the omelet making sure that it doesn't stick to the pan, gently slide in two spatulas (if the omelet is tearing at this point this means it's not ready, wait a little bit and cook it longer over very low heat) and quickly flip it over. Cook briefly on the other side, for about 30-60 seconds, or until the omelet is set. Transfer to a plate.
Whip the heavy whipping cream with the powdered sugar.
Serve the omelet with the whipped cream and fresh fruit.