Sweet Potato and Caramelized Onion Hash with Baked Eggs

Preheat oven to 450 °. Line a large baking sheet with foil or parchment paper. Melt butter in a cast-iron or other heavy skillet over medium-high heat. When it foams, add onions and sprinkle lightly with salt and pepper. (Don't worry if they are crammed into the pan; they will rapidly cook down.) Lower heat slightly and cook, stirring occasionally (reduce heat further if they seem to be burning), until dark brown, about 20-30 minutes. Meanwhile, toss sweet potatoes, oil, 1 1/2 tsp salt, smoked paprika, a generous helping of black pepper, and minced rosemary or oregano in a large bowl. Stir in onions. Spread sweet potato mixture on prepared baking sheet and roast for 25-40 minutes, stirring every 10-15 minutes, until sweet potatoes are soft and browned. Reduce oven temperature to 425 °. Spread a relatively thin layer of the cooked sweet potato hash in individual ramekins, a cast-iron skillet, or a 9" x 13" baking dish. Make small wells in the hash and crack in the eggs. Sprinkle lightly with salt and pepper. Bake for 10-20 minutes, or until hash is hot and eggs are baked to your preference. Test the eggs by prodding them with a fork to check the firmness of the white and the runniness of the yolk. Serve immediately, with Parmesan, if desired.