Chicken Hash with Eggs

Name: Chicken Hash with Eggs
Ingredients: 2 tablespoons unsalted butter; 2 tablespoons extra-virgin olive oil; 1 cup finely chopped green bell pepper; 3/4 cup finely chopped onion; 2 large jalapeños—stemmed, seeded and minced; 2 garlic cloves, minced; 1/2 teaspoon finely chopped thyme; Kosher salt; Freshly ground pepper; 1 large baked potato, peeled and cut into 1/2-inch dice (2 1/2 cups); 2 cups diced or shredded cooked chicken; 4 large eggs; Freshly grated Parmigiano-Reggiano cheese, for sprinkling