01 - Cook ramen noodles in boiling water according to package directions, omitting the seasoning packets. Drain thoroughly and allow to cool until just warm.
02 - In a mixing bowl, whisk eggs together with garlic powder, onion powder, salt, and pepper until fully blended.
03 - Add the drained noodles to the egg mixture and toss until noodles are evenly coated. Divide mixture into four portions and shape each into a flat, round bread-sized patty.
04 - Heat 1 teaspoon vegetable oil in a nonstick skillet over medium heat. Place two patties in the skillet and cook for 3 to 4 minutes on each side until set and golden brown. Transfer to a plate and repeat with remaining oil and patties. Let the patties cool slightly.
05 - Lay two noodle patties flat and evenly distribute cheddar and mozzarella over each. Place the remaining patties on top to assemble the sandwiches.
06 - Generously spread softened butter over the tops and bottoms of each assembled sandwich.
07 - Wipe the skillet clean and return to medium-low heat. Grill each sandwich for 2 to 3 minutes per side, pressing gently, until cheese is melted and the exterior is crisp and golden.
08 - Slice sandwiches in half and serve immediately while hot.