Cook the Ramen Noodles (1 package) 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 half of the Egg (2) 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 Canola Oil (2 tablespoon) 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.
Season the Beef Patty (1) with Soy Sauce (to taste), Salt and Pepper (to taste) and Sesame Oil (to taste). Wipe out the skillet, and cook the burger over medium-high heat until medium rare, or to your preferred degree of doneness.
Top with American Cheese Slice (1 slice).
Wipe out the skillet, and lower the heat to medium. Add 1 tablespoon of oil to the pan, and fry the other eggs sunny side up. For sriracha ketchup, whisk together the Sriracha (1/2 tablespoon) and Ketchup (1 tablespoon) in a small bowl.
Assemble the burger in this order, ramen bun, Arugula (1/2 cup), sriracha-ketchup, burger patty, fried eggs, thinly sliced Scallion (1) and top with the other ramen bun. Wrap in wax paper for easier eating, and serve hot.