Hawaiian Loco Moco {Hamburger Steak}

Name: Hawaiian Loco Moco {Hamburger Steak}
Prep time: 10 min
Cook time: 20 min
Total time: 30 min
Cuisine: American
Ingredients: 2 4-ounce ground beef patties; ⅛ teaspoon salt; ⅛ teaspoon ground black pepper; ⅛ teaspoon cayenne pepper (optional); 1 and ½ cups beef broth; 4 teaspoons low sodium soy sauce; 1 teaspoon Worcestershire (sauce); 4 teaspoons cornstarch; 2 teaspoon ketchup; 1 teaspoons unsalted butter (divided); 2 tablespoons green onions (minced, white & light green parts only); 2 large eggs; 2 cups cooked rice (I like white rice for this.)