Sweet Potato Hash With Caramelized Onions, Sausage, And Eggs

• 2 pound yellow onions
• 1 tablespoon unsalted butter
• 1 pound fresh italian sausage or chorizo
• 3 pound sweet potatoes
• 6 garlic cloves
• 2 sprigs fresh rosemary
• 3 tablespoon olive oil
• 1 tablespoon kosher salt, plus more to taste
• freshly ground black pepper
• 8 eggs
• parmigiano-reggiano cheese, to serve

1. Heat the oven to 450 degrees F.
2. Peel the onions and cut them into thin half-moons. Cut the half-moons in half.
3. Melt the butter in a skillet over medium-high heat. When it foams, add the onions and sprinkle with salt. Lower the heat and cook the onions for 30 minutes, stirring occasionally. Cook them until they are very dark brown and caramelized.
4. Put the sausage in another skillet and brown over medium-high heat, chopping it up into fine crumbles with a spatula. Cook the sausage for about 10 minutes, or until browned and beginning to crisp. Drain away any excess fat.
5. While the onions and sausage are cooking, chop the unpeeled sweet potatoes into ½-inch cubes. Finely mince the garlic and rosemary leaves, and toss them in a large bowl with the sweet potatoes. Toss with the olive oil, salt, and a generous helping of black pepper.
6. When the onions are dark brown and the sausage is beginning to get crispy, stir these into the sweet potatoes. Line a large baking sheet with foil, and spread out the sweet potato mixture evenly. Roast the sweet potatoes for 30-45 minutes, until they are soft and browned. If not serving immediately, cool the hash and refrigerate for up to 3 days.
7. Spread a thin layer of the sweet potato hash in a baking dish or cast iron skillet. Make small wells in the sweet potatoes and crack in large eggs. Sprinkle lightly with salt and pepper.
8. Bake for 10-20 minutes, until the sweet potatoes are hot and the eggs are cooked through. Serve immediately, with shavings of Parmigiano-Reggiano cheese, as desired.