Preheat oven to 375°F. Move the oven rack to the top position.
Rinse the large skillet and set it over medium heat. Add oil.
Once the oil is hot, add onion and peppers. Heat for 3-5 minutes, or until softened.
Add in garlic and cook for an additional 2-3 minutes.
Add the kielbasa and stir well.
Add paprika, salt, and pepper and stir, making sure everything is fully incorporated.
Add potatoes and stir. Cook 5-10 minutes, stirring every minute or so.
Sprinkle the cheese over the top of the hash.
Make three wells in the hash. (I used a soup ladle.) Crack an egg into each well, adding more salt and pepper, if desired.
Carefully place the hot skillet into the oven and bake for 20-23 minutes, rotating the skillet as needed every 5 minutes. This allows the eggs to cook more evenly. (For a runnier yolk, bake for a little less time; add more time for a firmer yolk.)
Carefully remove the skillet from the oven and garnish it with scallions and jalapeno. Add salt and pepper to taste.
Serve immediately.