Bring a saucepan of water to a boil and cook the ramen noodles according to the package directions, keeping them slightly firm. Drain, reserving a few tablespoons of the cooking water.

Melt the butter in a large skillet over medium heat. Add the minced garlic and red pepper flakes and cook for about 30 seconds, just until fragrant. Avoid browning the garlic.

Stir in the soy sauce and brown sugar. Cook for about 30 seconds, stirring until the sugar dissolves and the sauce becomes glossy.

Add the cooked ramen to the skillet and use tongs to toss until the noodles are evenly coated. Add the reserved noodle water and cook for another 1–2 minutes until the sauce clings to the noodles.

Push the noodles to one side of the skillet and pour the beaten eggs into the empty space. Scramble until just set, then toss the eggs into the noodles.

Turn off the heat and drizzle the sesame oil over the noodles. Toss gently once more so the flavors are evenly distributed.

Divide the ramen between two bowls and sprinkle with everything bagel seasoning and sliced green onions. Serve immediately while hot.