a recipe for madness

2 (3 ounce) packages instant ramen noodles, flavor packet discarded 2 large eggs salt and ground black pepper to taste 3/4 pound lean ground beef 1 tablespoon soy sauce 1 teaspoon sesame oil 3 tablespoons vegetable oil, divided 3 slices American cheese 1/4 cup ketchup 2 tablespoons chile-garlic sauce (such as Sriracha) 1 1/2 cups arugula 3 large eggs