Shred the cabbage, slice the green onions, julienne the carrots, and finely chop the mushrooms. Prepare the garlic and ginger. The more evenly everything is cut, the easier the egg rolls will be to fill and cook.
Heat 2 tablespoons neutral oil in a large skillet or wok over medium-high heat. Add the minced garlic and grated ginger and cook for 30–45 seconds, stirring constantly, until fragrant but not browned. Lower the heat if they begin to brown too quickly.
Add the shredded cabbage, carrots, mushrooms, and bean sprouts to the pan. Stir-fry for 3–5 minutes, until the cabbage starts to soften but still has some bite. Aim for tender-crisp vegetables so they keep good texture inside the egg rolls.
Add soy sauce, rice vinegar, and sesame oil, along with the salt and pepper. Sprinkle the cornstarch over the mixture and cook for another 1–2 minutes, stirring well, until excess moisture thickens slightly. Stir in the sliced green onions, then taste and adjust seasoning as needed.
Transfer the cooked filling to a large plate or bowl and let it cool to room temperature. You can chill it in the refrigerator for 10–15 minutes. Do not use hot filling, as it can steam the wrappers from the inside and make them soggy or prone to tearing.
On a clean surface, place an egg roll wrapper in front of you like a diamond (one corner pointing up). Keep a small bowl of water or beaten egg nearby for sealing, and line a tray or baking sheet with parchment paper for the finished rolls. Cover extra wrappers with a damp towel so they don’t dry out.
Spoon about 2–3 tablespoons of cooled filling just below the center of the wrapper, forming a short log that goes almost from side to side, leaving about 1 inch of space at each side. Avoid overfilling, which can cause the rolls to burst during frying.
Fold the bottom corner of the wrapper up over the filling, tucking it snugly underneath. Fold in the left and right corners toward the center like a burrito. Brush the top corner lightly with beaten egg or water, then roll up firmly, pressing gently as you go to push out any air. Press the edge to seal. Repeat with remaining wrappers and filling.
Pour enough neutral frying oil into a heavy-bottomed pot or Dutch oven to reach 2–3 inches up the sides. Heat over medium to medium-high heat until the oil reaches 350°F (175°C). If you don’t have a thermometer, dip a scrap of wrapper into the oil—if it bubbles quickly and rises to the surface, the oil is ready.
Working in small batches, gently lower 3–4 egg rolls into the hot oil using tongs, being careful not to crowd the pot. Fry for 3–5 minutes, turning occasionally, until the wrappers are deep golden brown and crispy. Adjust the heat as needed if they brown too quickly or too slowly.
Transfer fried egg rolls to a wire rack set over a baking sheet, or to a plate lined with paper towels. Let them rest for 3–5 minutes so the insides finish steaming and the outsides stay crisp.
Serve the vegetable egg rolls warm with sweet chili sauce, soy sauce, dumpling or gyoza sauce, or spicy mayo for dipping.