In a large skillet, heat 1 tablespoon oil over medium-high heat. Sauté the onions until soft, about 5 minutes. Add the rosemary and cook 1 minute more.

Add the potatoes and 1/3 cup water; reduce the heat to low and cook, covered, until tender, about 10 minutes.

Add the remaining 1 tablespoon oil, the chicken, and 1/4 teaspoon each of the salt and pepper to the skillet. Cook, turning only occasionally to allow the hash to brown nicely, until very dark golden all over, about 10 minutes. Transfer to a plate.

Heat the butter in the skillet. Crack the eggs into the pan and season with the remaining salt and pepper. Use a spatula to gently shape and lift the edges of the egg. Cook until the edges are brown and the egg centers softly set, about 5 minutes. Serve over the hash.