Heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease.
Heat vegetable oil in a separate skillet over medium heat.
Mix green onions, cabbage, carrot, garlic, 1 tablespoon sesame oil, 1 tablespoon sesame seeds, monosodium glutamate, salt, and pepper into ground beef mixture; cook and stir until liquid is evaporated and vegetables are tender, 5 to 10 minutes. Transfer beef mixture to a bowl and mix in 1 egg.
Crack the second egg into a bowl and beat well.
Hold 1 wonton wrapper in the palm of your hand and brush a thin layer of beaten egg on 1 edge. Scoop about 1 teaspoon beef mixture into the center of the wrapper. Fold wrapper in half, corner to corner, to make a triangle and pinch the edges shut, crimping with your fingers to make a seal. Press the air out by cupping your fingers over the dumpling in your palm and pressing lightly.
Fry wontons in the hot oil until 1 side is browned, 2 to 3 minutes. Flip and cook until other side is browned, 2 to 3 minutes. Transfer wontons to a paper towel-lined plate to drain using a slotted spoon.
Whisk soy sauce, rice wine vinegar, 1 teaspoon sesame oil, and 1 teaspoon sesame seeds together in a bowl until dipping sauce is smooth. Serve alongside wontons.
