Makes six 3-egg omelets
1 teaspoon Dijon mustard
1 to 2 tablespoons olive oil
A heaping 1/4 cup grated Gruyère cheese
To prepare the filling, melt the butter in a large heavy skillet over medium-high heat. Add the onions and thyme and reduce the heat to medium-low. Cover and cook, stirring often, until the onions are very soft, about 20 minutes. Uncover and increase the heat to medium-high. Cook, stirring often, until the onions are light brown, about 20 minutes. Add the stock and wine. Season with salt and pepper. Simmer until the mixture is thick and pulpy, 10 to 15 minutes. Stir in the mustard and taste and season with additional salt and pepper if desired. You should have about 1 1/2 cups filling.For the croutons, preheat the oven to 400°F and cut the bread into 1/3-inch dice. Toss the cubes with the olive oil until well coated and place on a baking sheet. Bake until golden brown, 8 to 10 minutes. Let cool.
For each omelet, break the eggs into a small bowl and add 2 teaspoons water and a pinch of salt and pepper. Beat with a fork until the whites and yolks are well blended. Heat 1 tablespoon butter in a small omelet pan or nonstick skillet over high heat until foaming and just starting to brown. Have a warmed plate ready to receive the omelet. Add the eggs and stir quickly with the flat part of a fork until they begin to thicken, about 10 seconds. Push the egg that begins to set on the sides into the center of the pan and tilt the pan to pour the uncooked egg to the sides. Cook until the omelet is slightly firm, 15 to 20 seconds longer.
Stop stirring and let the omelet brown slightly on the bottom. Sprinkle a heaping 1/4 cup cheese over the surface. Spread about 1/4 cup of the onion mixture on the cheese and add 1/4 cup croutons. Fold the sides of the omelet over the filling and tilt the pan to half roll, half slide it onto the plate, seam side down. Serve at once.