How to Make It
Melt the butter in a large skillet over medium heat. Add the potatoes and onion and cook, stirring occasionally, until they begin to soften, 4 to 5 minutes.
Stir in half the parsley, 1 teaspoon salt, and ½ teaspoon pepper. Using a spatula, spread the potatoes, pressing into the skillet. Cook until a crust forms, 4 to 6 minutes.
Break the potatoes apart, flip, and press again. Cook until another crust forms, 4 to 6 minutes.
Reduce heat to medium-low, cover, and cook until tender and crisp, 8 to 10 minutes.
Meanwhile, heat the oil in a second large skillet over medium heat. Prick the sausages with a fork and cook, turning occasionally, until golden brown and cooked through, 15 to 20 minutes. Transfer to plates.
Pour off all but about 2 tablespoons of the fat from the second skillet. Add the tomatoes and ½ teaspoon each salt and pepper. Cook, tossing occasionally, until the tomatoes begin to soften, 2 to 3 minutes. Stir in the remaining parsley. Serve with the sausages and hash browns.