Arrange a rack in the middle of the oven and heat to 450°F.
Melt 1 tbsp of the butter in a large skillet over medium-high heat. Add the onions and sprinkle lightly with 1 tsp of the salt. Lower the heat to medium and cook, stirring occasionally for 30 to 40 minutes.
Place the sausage in another skillet and brown over medium-high heat, chopping it up into fine crumbles with a spatula. Cook the sausage until browned and beginning to crisp, about 10 minutes. Drain away any excess fat; set aside.
Line a rimmed baking sheet with aluminum foil or parchment paper. Place the sweet potatoes, onions, sausage, salt, melted butter, garlic, rosemary, and a generous helping of black pepper in a large bowl and toss to combine. Transfer to the baking sheet and spread into an even layer.
Roast until the sweet potatoes are soft and browned, 30 to 45 minutes
Lower the oven temperature to 425°F. Using the back of a large spoon, make eight wells in the hash. Crack an egg into each well. Season the eggs with salt and pepper. Bake until the eggs are done to your liking, 8 to 10 minutes.
Serve immediately, with shavings or sprinkles of Parmesan cheese, if desired.