In a large nonstick skillet, cook the sausage, breaking it up with a wooden spoon, over medium-high heat until browned, about 5 minutes; remove from the heat.
In a medium cast-iron skillet, heat the olive oil over medium heat. Add the frozen potatoes, season with 1/4 teaspoon each salt and pepper and cook, turning infrequently, until browned, about 15 minutes.
Meanwhile, in a small saucepan, melt 1 tablespoon butter over medium heat. Reserve half of the scallion greens, then add the remaining scallions, 1/4 teaspoon salt and a pinch pepper and cook, stirring, until softened, about 2 minutes. Stir in the flour and cook for 1 minute. Gradually stir in the milk and cook, stirring, until boiling and thickened, about 2 minutes. Remove from the heat and cover to keep warm.
Add the remaining 1 tablespoon butter to the nonstick skillet with the sausage and melt over medium-high heat. Add the tomatoes, the beaten whole eggs and egg whites and a couple pinches salt and pepper and cook, scrambling, until just set, 2 to 3 minutes. Divide the egg mixture among 4 plates; top with the hash browns, white sauce and cheddar. Garnish with the reserved scallion greens.