Primavera Risotto Nests with Fried Eggs

Name: Primavera Risotto Nests with Fried Eggs
Ingredients: 4 tablespoons (1/2 stick) butter, divided; 2 cups chopped button mushrooms (about 5 ounces); 3/4 cup chopped onion; 3 garlic cloves, minced; 2 cups arborio rice; 1/2 cup dry white wine; 3 cups vegetable broth, divided; 3/4 cup 1/3-inch cubes carrots; 2 cups diced trimmed asparagus (about 9 ounces); 3 cups (or more) water; 1 cup freshly grated Parmesan cheese plus additional for serving; 1/2 cup shelled fresh peas or thawed frozen peas; 1/4 cup chopped fresh Italian parsley; 1 tablespoon olive oil plus additional for drizzling; 4 large eggs