1. Heat 1 tablespoon of the olive oil in a large heavy skillet over medium-high heat. Add the breakfast sausage and cook, breaking it up with a spatula, until browned and cooked through, 5–7 minutes. Transfer the sausage to a plate, leaving the drippings in the pan.

2. Add the remaining 1 tablespoon olive oil to the skillet. Stir in the diced potatoes and spread them into an even layer. Cook without stirring for 4–5 minutes, until the bottoms are deeply golden and starting to crisp.

3. Stir the potatoes, then add the sliced onion, diced red bell pepper, and minced garlic. Sprinkle with the smoked paprika, dried oregano, crushed red pepper flakes (if using), 3/4 teaspoon of the kosher salt, and 1/2 teaspoon of the black pepper. Cook, stirring occasionally, until the potatoes are tender and the onions are lightly caramelized, 8–10 minutes.

4. Return the cooked sausage to the skillet and stir to combine with the potatoes and vegetables. Taste and adjust the seasoning if needed, then use the back of a spoon to make 4 small wells in the hash.

5. Crack one egg into each well. Sprinkle the eggs with the remaining 1/4 teaspoon kosher salt and 1/4 teaspoon black pepper. Reduce the heat to medium-low, cover the skillet, and cook until the egg whites are set and the yolks are cooked to your liking, about 4–6 minutes.

6. Remove the skillet from the heat and sprinkle the top with the 2 tablespoons chopped fresh parsley. Serve straight from the pan so everyone gets an egg and a generous scoop of the sausage and potato hash.