Preheat the oven to 400˚F.
In a medium skillet over medium high heat, warm oil. Once hot, add potatoes, bell pepper, onion, and jalapeno. Cook for about 7 to 8 minutes, stirring frequently, until the vegetables are softened and you can pierce the potato with a fork.
Stir in ham, garlic, cumin, smoked paprika, pepper, salt, and chili powder. Cook for 1 to 2 minutes more, stirring frequently, until the garlic is fragrant.
Remove from heat. Use a spoon to make 4 wells (you can also make 6 and use 6 eggs if you prefer!) in the hash mixture. Add eggs to the wells and sprinkle cheese over the vegetables.
Bake for 8 to 10 minutes, until the egg whites are set but the yolks are still runny. Remove from the oven.
Garnish with freshly ground black pepper and chopped cilantro. Serve immediately.