Cook 1/2 cup sliced cremini mushrooms in 2 teaspoons extra-virgin olive oil in an 8-inch nonstick skillet over medium heat until caramelized, 6 to 8 minutes. Season with salt and pepper; transfer to a bowl. Stir 1/4 cup packed watercress into mushrooms until slightly wilted. Return skillet to heat.
Using a fork (for good control), beat eggs with 1 teaspoon water until yolks and whites are completely combined and no streaks remain. The water helps incorporate them and steams while the omelet cooks, which puffs it up. Season with 1/8 teaspoon kosher salt and a few cracks of pepper.
Heat butter over medium-high heat until foaming; swirl to coat. When foam subsides, pour eggs into center of skillet. Let stand 10 seconds. Stir eggs with a rubber spatula in a circular motion with one hand while shaking skillet with the other, about 15 seconds. Reduce heat to medium. Working your way around skillet, gather outer set eggs toward center, then tip skillet so runny eggs move to outside edges, about 1 minute. Lightly spread any remaining runny eggs evenly over top and cook until creamy but no longer wet.
Add mushroom-watercress mixture and 2 tablespoons finely grated Parmigiano-Reggiano onto far side of omelet. Remove from heat. Tip pan away from you; gently slide spatula under near side of omelet and fold up and over cheese. Slide onto a warm plate. Top with more finely grated Parmigiano-Reggiano or chopped chives and serve with buttered toast, if desired.