Heat 1 tablespoon of the oil in a large ovenproof nonstick skillet over medium heat. Add shallot and cook, stirring often, until translucent, about 3 minutes. Add mushrooms and increase heat to medium-high. Cook mushrooms until they release their liquid and brown lightly, about 7 minutes. Add Swiss chard leaves, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper; cook until leaves are wilted, about 1 minute. Remove from heat; transfer mixture to a bowl and wipe skillet clean with paper towels.
Beat egg whites with an electric mixer on high speed until stiff peaks form, 3 to 4 minutes. Whisk together egg yolks and remaining 1/4 teaspoon each salt and pepper in a separate bowl. Using a spatula, carefully fold egg whites into yolks just until combined.
Heat remaining 1 tablespoon oil in skillet over medium heat. Spread egg mixture evenly in skillet. Cook until a crust begins to form, about 30 seconds. Sprinkle with vegetable mixture and cheese.
Bake until omelet has risen and looks puffy, about 3 minutes. Using a spatula, carefully transfer to a cutting board. Sprinkle with basil, parsley, and chives and cut into 8 wedges. Serve immediately.
8 ounces Swiss chard
2 tablespoons extra-virgin olive oil
2 tablespoons finely chopped shallot (from about 1 small shallot)