1 package ramen noodles
2 tablespoons neutral oil, such as canola or grapeseed
1 tablespoon ketchup
1/2 tablespoon sriracha
1 beef burger patty
1 slice American cheese
1 scallion, thinly sliced on the bias
1/2 cup arugula
Cook the ramen according to the directions on the package (including the seasoning), then drain off the liquid. Let cool to room temperature. In a small mixing bowl, whisk 1 egg until no streaks of yolk remain. Add the ramen, tossing thoroughly to coat with the egg. Divide the egg-dressed ramen into two portions, and place each half into a ramekin.
Cover the ramen with plastic wrap, and weigh it down with a can of soup to compress it into a bun shape. Refrigerate for at least 15 minutes.
Add the oil to a skillet, and set over medium-high heat. Once the oil begins to shimmer, unmold the ramen buns into the pan, and cook until the bottom is a dark golden brown. Flip both of the patties, and cook until a dark golden brown on both sides.
Whisk together the sriracha and ketchup in a small bowl.
Season the hamburger patty with salt, pepper, a splash of soy sauce, and sesame oil. Wipe out the skillet, and cook the burger over medium-high heat until medium rare, or to your preferred degree of doneness. Finish with a splash of soy sauce and sesame oil. Top with a slice of cheese.
Assemble the burger in this order: ramen bun, arugula, ketchup, burger patty (cheese side up), scallions, and the second ramen bun. Wrap in wax paper for easier eating, and serve hot.