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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.
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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.
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3
Mix beef, soy sauce, and sesame oil in a bowl. Divide beef mixture into three equal portions and shape into patties.
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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.
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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.
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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.
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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.
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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.
