Step 1
Bring a pot of water to a boil; add ramen noodles. Boil stirring occasionally until noodles are tender about 3 minutes. Drain. Allow noodles to cool slightly.
Step 2
Beat 2 eggs in a bowl; season with salt and pepper. Stir noodles into eggs until evenly coated. Divide noodles into 6 burger-size ramekins or bowls. Place a sheet of plastic wrap directly over noodles and stack another bowl on top to flatten noodles. Refrigerate until firm about 20 minutes.
Step 3
Mix beef soy sauce and sesame oil in a bowl. Divide beef mixture into three equal portions and shape into patties.
Step 4
Heat about 1 tablespoon vegetable oil in a large skillet over medium-high heat. Invert each ramen bowl gently tapping the bottom of the bowl until ramen loosens careful to maintain bun shape.
Step 5
Fry each ramen bun without moving noodles in hot oil until golden brown about 3 minutes. Flip each bun and continue frying until crisp 3 to 5 minutes more. Ramen bun should be crispy on one side and slightly softer on the other. Transfer buns to large plate with the crispy side up.
Step 6
Heat about 1 tablespoon vegetable oil in the same skillet used to fry ramen over medium heat. Cook beef patties until burgers are lightly pink in the center 3 to 5 minute per side. An instant-read thermometer inserted into the center should read 140 degrees F (60 degrees C). Arrange a slice of American cheese over each burger and continue to cook until melted about 1 minute more.
Step 7
Mix ketchup and chile-garlic sauce in a small bowl; spread mixture over the crispy side of each ramen bun. Divide arugula over 3 ramen buns. Place burgers over arugula.
Step 8
Heat about 1 teaspoon oil in a small skillet over medium-high heat; fry remaining eggs one at a time adding more oil as needed until yolk is almost firm about 1 minutes per side. Place egg over patty and top with remaining bun.