In a large heavy skillet set over medium-high heat, heat 1 tablespoon of the olive oil. Add the sausage and cook, breaking it up with a wooden spoon, until browned, 5 minutes. Transfer it to a paper towel-lined plate.

Heat the remaining 2 tablespoons olive oil in the same skillet over medium-high heat. Add the potatoes, season liberally with salt and pepper, and sauté until golden brown, 10 to 12 minutes. Add the onion, bell peppers, and jalapeno, and cook for 5 more minutes. Stir in the cooked sausage and the cheese, and season to taste with salt and pepper.

Crack the eggs onto the hash in the skillet. Cover with a lid and cook until eggs are set, 8 to 10 minutes. Divide the hash and eggs among four plates and serve.