To preserve the puff of a puffy omelet, fold the omelet over its filling or serve it open-faced with the filling on top as soon as you remove it from the oven.
In large mixing bowl at high speed, beat egg whites with water and cream of tartar until stiff but not dry, just until whites no longer slip when bowl is tilted. In small mixing bowl at high speed, beat egg yolks with salt, if desired, until thick and lemon-colored Gently, but thoroughly, fold yolks into whites.
In 10-inch omelet pan with ovenproof handle over medium-high heat, heat butter until just hot enough to sizzle a drop of water.Pour in egg mixture and gently smooth surface, Reduce heat to medium. Cook until puffed and lightly browned on bottom, about 5 minutes. (Lift omelet at edge to judge color.) Bake in preheated 350ºF oven until knife inserted halfway between center and outer edge comes out clean, about 10 to 12 minutes. Loosen omelet edges with spatula.
To serve folded: With sharp knife, cut upper surface down center of omelet but DO NOT cut through to bottom of omelet. Fill, if desired. Tip pan. With pancake turner, fold omelet in half and invert onto warmed plate or platter with a quick flip of the wrist.
To serve open-faced: Invert pan over warmed plate or platter, or slide omelet from pan onto plate. Spoon or sprinkle filling, if desired, over top.
Cut in half or into wedges. Serve immediately.