Cover mushrooms with 1 1/4 cups hot water in a medium bowl. Set a small bowl on top of mushrooms to keep them submerged. Let stand until soft, about 20 minutes. Remove mushrooms with a slotted spoon. Cut large mushrooms lengthwise into 1/4-inch strips; keep small mushrooms whole. Set aside. Strain soaking liquid; set aside.

Melt 1 teaspoon butter in a medium nonstick skillet over medium-high heat. Add scallions and 1/4 teaspoon salt. Cook, stirring, until scallions are wilted, about 2 minutes. Stir in mustard, mushrooms, and 1/4 cup soaking liquid. Cover; cook, stirring, until mushrooms are tender and liquid has evaporated, about 3 minutes more. Set aside.

Whisk eggs, egg whites, remaining salt, and 2 pinches of pepper in a medium bowl. Melt 1/2 teaspoon butter in a small nonstick skillet over medium heat. Add half of the egg mixture. Stir gently with a rubber spatula to create curds. Use spatula to pull cooked egg awayfrom sides, allowing raw egg to flow underneath; cook until almost set, 3 to 4 minutes. Top with half of the mushroom filling. Cook until set, 1 to 2 minutes more. Gently fold one half over the other. Slide omelet onto a plate. Repeat with remaining butter, eggs, and filling. Season with pepper.