Heat 1 1/2 teaspoons oil in a skillet over medium-high heat. Add cabbage, bean sprouts, and carrots; saute until tender, 6 to 8 minutes. Transfer vegetable mixture to a large bowl.
Place beef in the same skillet; cook and stir until no longer pink, 5 to 7 minutes. Drain beef into the bowl with the vegetable mixture. Add green onions, beaten egg, garlic, sesame oil, sesame seeds, salt, pepper, and ginger. Mix filling well.
Separate and place wonton wrappers onto a flat work surface. Spoon 1 tablespoon of the filling onto the center of each wrapper. Use your finger or a pastry brush to lightly moisten edges of wonton wrappers with water. Fold one corner of the wrapper over the filling onto the opposite corner to form a triangle. Press edges together to seal wontons.
Heat remaining 2 tablespoons oil in a large saucepan to 375 degrees F (190 degrees C). Place wontons in the hot oil; fry until lightly browned, 1 to 2 minutes per side.