Heat 2 tablespoons olive oil in a large cast-iron skillet over medium-high heat. Add diced potatoes in an even layer and cook for 12-15 minutes, stirring occasionally, until golden brown and crispy.
Push potatoes to the sides of the skillet and add sausage to the center. Break up with a spoon and cook for 5-6 minutes until browned and cooked through.
Add remaining 1 tablespoon oil, then stir in onion and bell pepper. Cook for 4-5 minutes until softened. Add garlic, paprika, salt, and pepper, cooking for 1 more minute until fragrant.
Create four wells in the hash mixture and crack an egg into each well. Cover the skillet and cook for 4-6 minutes until egg whites are set but yolks are still runny (or cook longer if you prefer hard yolks).
Remove from heat and sprinkle with shredded cheddar cheese and fresh parsley. Serve immediately straight from the skillet for that cozy family-style breakfast!