Chef John's Loco Moco

Name: Chef John's Loco Moco
Prep time: 15 min
Cook time: 15 min
Total time: 30 min
Cuisine: American, Asian
Ingredients: 2 (4 ounce) patties ground beef; 1 pinch salt and freshly ground black pepper to taste; 1 pinch cayenne pepper; 1.5 cups beef stock; 4 teaspoons soy sauce; 1 teaspoon Worcestershire sauce; 4 teaspoons cornstarch; 2 teaspoons ketchup; 2 teaspoons white sugar; 2 drops sesame oil, or to taste; 2 teaspoons unsalted butter, divided; 2 tablespoons minced green onions, white and light green parts only; 2 eggs; 2 cups hot cooked white rice