Preheat the oven to 375 degrees F. Put the broccoli in a microwave-safe bowl and sprinkle with 3 tablespoons water. Cover with plastic wrap and microwave until crisp-tender, about 4 minutes; drain. Meanwhile, heat the butter in a medium nonstick ovenproof skillet over medium heat. Add the hash browns and season lightly with salt and pepper. Cook, stirring occasionally, until golden brown, about 15 minutes. Whisk the eggs, milk, cheese, 1/2 teaspoon salt, and pepper to taste in a large bowl and stir in the broccoli. Pat the hash browns into an even layer, then pour the egg-broccoli mixture on top. Transfer the skillet to the oven and bake until the omelet is just set, 15 to 20 minutes. Whisk the olive oil, vinegar and mustard in a bowl. Add the tomatoes and scallions and toss. Season with salt and pepper. Invert the omelet onto a plate and cut into wedges. Serve with the tomato salad.