Heat 2 tablespoons of the oil in a large skillet over medium heat. Add the sliced sausage and cook for 5 to 6 minutes until the edges brown. Remove the sausage to a plate and leave the drippings in the pan.
Add the remaining tablespoon of oil to the skillet along with the diced potatoes. Spread them in an even layer and let them cook undisturbed for 8 minutes so the bottoms crisp.
Stir in the sliced onions, smoked paprika, garlic powder, salt, and pepper. Continue cooking for 10 to 12 minutes, stirring every few minutes, until the potatoes are tender and the onions soften and turn golden.
Return the browned sausage to the skillet and toss everything together. Cook for 2 more minutes to warm the sausage through.
In a separate small skillet or by pushing the hash to one side of the large skillet, fry the eggs in a little extra oil or butter until the whites set but the yolks stay runny.
Divide the hash among four plates and top each serving with a fried egg.