Beat the eggs in a large bowl until foamy. Heat the olive oil in a large nonstick ovenproof skillet over medium-high heat. Add the ham, bell peppers, onion, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the vegetables are tender and starting to brown, about 6 minutes; add to the bowl with the eggs.
Preheat the broiler. Wipe out the skillet and return to medium-high heat. Melt the butter in the skillet, then add the frozen hash browns in an even layer. Cook, pressing the hash browns into the bottom and up the side of the skillet with a spatula, until golden on the bottom, about 6 minutes.
Pour the egg mixture over the hash browns. Transfer the skillet to the broiler; cook until partially set, 3 to 5 minutes. Sprinkle the cheese and all but about 2 tablespoons scallions over the eggs. Continue broiling until the cheese melts and the eggs are set, 2 to 3 more minutes.
Loosen the edge of the omelet with a small spatula. Slide onto a platter and fold in half. Cut into wedges; top with the reserved scallions.