In a bowl, whisk together the eggs, 2 tablespoons water, ½ teaspoon salt, and ¼ teaspoon pepper.
Melt the butter in a large nonstick skillet over medium heat. Cook the eggs, without stirring, until they begin to set, 2 to 3 minutes.
With a rubber spatula, lift the edges of the omelet, tilting the pan so the uncooked eggs flow to the edges of the pan. Cook until set, 1 minute.
Sprinkle the cheese and scallions over the eggs. Fold a third of the omelet over the center; fold over the other third. In the pan, cut into 4 pieces.
Whisk together the oil, vinegar, mustard, and ¼ teaspoon each salt and pepper. Drizzle over the arugula. Serve with the omelet and bread.