Heat a large skillet over medium-high heat. Add the ground beef and cook until deeply browned, breaking it apart with a wooden spoon as it sears. Drain any excess grease from the pan.

Lower the heat to medium and stir in the minced garlic and grated ginger, cooking for about 1 minute until fragrant. Add the shredded cabbage and carrots to the skillet. Toss everything together and cook for 3 to 4 minutes, just until the vegetables begin to wilt.

Stir in the soy sauce and toasted sesame oil, tossing to coat the beef and vegetables evenly. Remove the skillet from the heat and transfer the filling to a colander set over a bowl. Let the filling drain and cool for at least 15 minutes.

Place an egg roll wrapper on a clean surface in a diamond shape. Spoon about 3 tablespoons (45g) of the cooled filling onto the bottom third. Fold the bottom corner up and tightly over the filling, fold in the two side corners, and roll tightly toward the top point. Brush the top corner with the beaten egg and press firmly to seal.

Pour the vegetable oil into a deep Dutch oven and heat to 350°F (175°C). Carefully lower 3 to 4 egg rolls into the hot oil. Fry for 4 to 5 minutes, turning occasionally, until the wrappers are beautifully blistered and a deep golden brown.

Use a spider strainer to carefully remove the egg rolls. Place them on a wire rack set over a baking sheet to drain. Serve immediately.