In a wide skillet, warm a tablespoon of oil over medium-high heat. Crumble in the pork and cook until it turns golden and no longer pink, breaking it apart with a spoon. Discard any rendered fat.
Stir in the cabbage, carrots, chopped green onions, and garlic. Sauté the mixture for about 3–4 minutes until the vegetables are slightly softened.
Incorporate the soy sauce, oyster sauce, sesame oil, and ground ginger into the skillet. Season the mixture with salt and freshly ground pepper to your preference. Continue cooking for an additional 2 minutes, then remove from heat. Let the filling mixture cool completely.
Position an egg roll wrapper on a clean surface so that it resembles a diamond shape. Spoon roughly 2 tablespoons of the cooled filling into the center.
Fold the bottom point of the wrapper over the filling, then fold the left and right corners inward over the filling. Brush the top corner lightly with the beaten egg, then roll upwards tightly to seal the egg roll.
In a deep pan or wok, heat about 2–3 inches of vegetable oil to a steady 350°F (175°C). Carefully lower the egg rolls in small batches into the hot oil and fry each for 2–3 minutes per side until they turn crisp and golden brown.
Transfer the cooked rolls to a paper towel-lined plate to drain any excess oil.
Serve the egg rolls warm, accompanied by sweet and sour sauce, soy sauce, or hot mustard, according to your preference.