8 or more large eggs

Kosher salt

Freshly ground black pepper

Parmesan cheese (optional)

To make the hash:

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.

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, 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.

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.

To continue cooking immediately:

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