Instructions

1. To make the hash:
2. Arrange a rack in the middle of the oven and heat to 450°F. Peel the onions and cut them in half lengthwise, then cut them into thin half-moons. Cut the half-moons in half. Melt 1 tablespoon of the butter, ghee, or olive oil in a large skillet over medium-high heat. Add the onions and sprinkle lightly with 1 teaspoon of the salt. (Don't worry if they are crammed into the pan; they will rapidly cook down.) Lower the heat to medium and cook, stirring occasionally and lowering the heat further if they seem to be burning, until they are very dark brown and caramelized, 30 to 40 minutes. Meanwhile, cook the sausage.
3. 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.
4. Line a rimmed baking sheet with aluminum foil or parchment paper. Place the sweet potatoes, onions, sausage, remaining 2 teaspoons salt, remaining 3 tablespoons of melted butter or oil, 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.
5. Roast until the sweet potatoes are soft and browned, 30 to 45 minutes (roasting time depends on the size and uniformity of the sweet potato chunks, as well as the variety of sweet potato you buy). After this, you can either refrigerate the hash in an airtight container overnight or finish the hash now.
6. To continue cooking immediately:
7. To continue cooking immediately, lower the oven temperature to 425°F. Using the back of a large spoon, make 8 eight wells (or however many eggs you desire) 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. Test the eggs by prodding them with a fork to check the firmness of the white and the yolk; baked eggs are deceptive in that the white often looks much less cooked than it really is.
8. Serve immediately, with shavings or sprinkles of Parmesan cheese, if desired.
9. To serve later:
10. To serve later, when you are ready to cook, arrange a rack in the middle of the oven and heat to 425°F. Spread a relatively thin layer of the sweet potato hash in a baking dish, such as a cast iron skillet or a 13x9-inch baking dish. You can also bake in individual ramekins. Make small wells in the sweet potatoes and crack an egg into each well. Season the eggs with salt and pepper.
11. Bake until the sweet potatoes are hot and the eggs are baked through, 10 to 20 minutes, testing the eggs by prodding them with a fork to check the firmness of the white and the yolk (baked eggs are deceptive in that the white often looks much less cooked than it really is.)
12. Serve immediately, with shavings or sprinkles of Parmesan cheese, if desired.