Cook ramen according to directions on packet without adding the spice packet. Drain, rinse and allow to cool.
Beat two eggs in a bowl and season with salt and pepper. Stir cooled noodles into the eggs until evenly coated. Divide noodles into six burger-sized bowls or baking dishes. Cover and press until flat. Refrigerate until firm, for about 20 minutes.
Combine beef, soy sauce, and sesame oil in a large bowl. Divide mixture into three equal patties
Heat one tablespoon olive oil in a large skillet over medium-high heat. Carefully remove each ramen patty by flipping the bowls over and gently tapping until the ramen loosens.
Fry each ramen patty in oil for about three minutes per side, or until crispy and golden brown. Transfer buns to a paper towel-lined plate
Heat one tablespoon olive oil in the same skillet. Cook beef patties according to how you like it, about three to five minutes per side for medium-well. Place cheddar cheese slices on patties, cover, and continue to cook for one more minute, or until cheese is melted.
Place burgers on ramen buns, top with lettuce and tomato slices, and serve.