Heat a large skillet over medium-high heat. Add the ground pork or chicken and cook until thoroughly browned, breaking it into small pieces with a wooden spoon. Add the minced garlic and grated ginger, then cook for about 30 seconds, until fragrant.
Stir in the shredded green cabbage, carrots, and green onions. Cook until the vegetables have softened and the excess moisture has evaporated, about 4–5 minutes.
Add the soy sauce, oyster sauce, sesame oil, sugar, white pepper, and cornstarch. Stir everything together until well combined. Remove the skillet from the heat and let the filling cool completely.
Place an egg roll wrapper on a clean, dry surface with one corner pointing toward you. Spoon 2–3 tablespoons of the cooled filling near the bottom corner of the diamond.
Roll the Egg Rolls: Fold the bottom corner tightly over the filling, then fold in both side corners. Roll the wrapper firmly upward toward the top point, then seal the top edge with the prepared egg wash. Repeat until all the filling has been used.
Heat the vegetable or peanut oil in a heavy pot or Dutch oven to exactly 350°F (175°C). Fry 3–4 egg rolls at a time for 3–4 minutes, turning them occasionally, until they are beautifully golden brown.
Carefully transfer the hot, fried egg rolls to a wire cooling rack set over paper towels to drain the excess oil.
Serve the egg rolls piping hot with a side of sweet chili sauce, duck sauce, or extra soy sauce for dipping.