Heat 1 tablespoon vegetable oil in a large skillet over medium-high heat. Add ground pork, breaking it up with a spatula. Cook until no longer pink and slightly browned, about 4 to 6 minutes. Drain excess fat if needed.

Push meat to the side, add garlic and ginger, and cook until fragrant, about 30 seconds. Add the shredded cabbage and grated carrot and stir-fry until wilted but still slightly crisp, 2 to 3 minutes. Stir in bean sprouts and the white parts of green onions. The filling should be glossy but not watery; if there is excess liquid, cook another minute until it evaporates.

Stir in soy sauce, oyster sauce, sesame oil, and pepper. Taste and adjust seasoning. Transfer filling to a bowl and let it cool for 10 minutes.

Lay one egg roll wrapper with a corner facing you. Spoon about 2 to 3 tablespoons of filling near the bottom corner. Fold the bottom corner up over the filling, fold both side corners in snugly, then roll tightly toward the top corner. Brush the top corner with the egg wash and press to seal. Repeat with remaining wrappers and filling.

Heat oil in a deep pot or wok to 350 to 375°F (175 to 190°C). Carefully add 3 to 4 egg rolls without overcrowding. Fry, turning occasionally, until evenly golden brown and blistered, about 6 to 8 minutes total.

To bake, preheat oven to 400°F (200°C). Brush each roll lightly with oil and place seam side down on a parchment-lined baking sheet. Bake for 15 to 20 minutes, turning halfway, until golden and crisp.

To air fry, cook at 375°F (190°C) for 8 to 12 minutes, turning once.

Let egg rolls rest 2 minutes before serving so the filling settles. Serve with dipping sauces of your choice.