How to Make It
In a large cast-iron skillet, melt the butter in the olive oil. Add the green pepper, onion, jalapeños, garlic, thyme and a generous pinch each of salt and pepper. Cook over moderate heat, stirring occasionally, until softened and just starting to brown, about 7 minutes. Stir in the potato and chicken and flatten the mixture with a spatula. Cook over high heat until crusty on the bottom, about 5 minutes. Working in sections, turn the hash over and cook until browned on the other side, 3 to 5 minutes.
Using the back of a spoon, make 4 depressions in the hash. Carefully crack the eggs into the depressions and season with salt and pepper. Cover the skillet and cook over moderate heat until the egg whites are just set but the yolks are runny, about 4 minutes. Sprinkle the hash with grated cheese and serve right out of the skillet.