Heat a large skillet over medium-high heat with a tiny drizzle of oil. Add minced garlic and grated ginger and cook for 30 seconds until fragrant. Add the chopped shrimp and cook for 2 minutes until just barely pink. Remove from heat — do not fully cook through.

Add shredded cabbage, carrots, and green onions to the pan. Stir into the shrimp mixture and let sit in residual heat for 1 minute until cabbage is slightly wilted. Add soy sauce, sesame oil, oyster sauce, black pepper, and cornstarch. Mix well. Spread filling on a plate and cool completely before rolling.

Lay an egg roll wrapper flat on a clean surface in a diamond orientation. Spoon 2 to 3 tablespoons of cooled filling into the center.

Fold the bottom corner up over the filling, fold in both side corners, then roll forward tightly. Brush the top corner with beaten egg and press firmly to seal. Repeat with remaining wrappers and filling.

Heat vegetable oil in a heavy skillet to approximately 350°F — about 1 inch of oil depth. Test readiness by dropping in a small piece of wrapper; it should sizzle immediately and float.

Fry egg rolls in batches of 3 to 4, turning occasionally, for 3 to 4 minutes until deep golden brown all over. Do not crowd the pan.

Transfer to a paper towel-lined plate. Rest for 1 minute before serving. Serve with sweet chili sauce or soy dipping sauce.