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.
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.
Mix beef, soy sauce, and sesame oil in a bowl. Divide beef mixture into three equal portions and shape into patties.
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.
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.
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.
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.
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.