Finely shred the green and purple cabbage. Grate or julienne the carrots. Rinse and drain the bean sprouts very well, then pat dry with a clean kitchen towel if they seem wet to avoid a soggy filling. Finely chop the mushrooms, slice the green onions, and mince the garlic and ginger.

In a large skillet or wok, heat 2 tablespoons neutral oil over medium-high heat. Add the minced garlic and ginger and cook for 30–45 seconds, stirring constantly, until fragrant but not browned.

Add the green cabbage, purple cabbage, carrots, mushrooms, and bean sprouts to the pan. Stir-fry for 3–5 minutes, until the cabbage just begins to wilt but is still tender-crisp. Add green onions during the last minute of cooking.

Reduce heat to medium. Add soy sauce, rice vinegar, toasted sesame oil, ground white pepper, and salt. Stir well to coat the vegetables evenly. Taste carefully and adjust seasoning with a bit more soy or vinegar if needed. If using hoisin or chili garlic sauce, stir them in now along with any optional napa cabbage or bell pepper and cook 1–2 minutes more.

Transfer the cooked vegetable mixture to a large bowl or spread it on a sheet pan. Let it cool to room temperature for at least 15–20 minutes. Cooling prevents steam from making the wrappers soggy.

Stack the egg roll wrappers and keep them covered with a slightly damp towel so they don’t dry out. Place a small bowl of beaten egg (or water/plant milk) nearby for sealing. Line a baking sheet with parchment paper to hold the rolled egg rolls.

Working with one wrapper at a time, place it on a clean surface in a diamond shape (one point facing you). Spoon about 2–3 tablespoons of cooled filling just below the center. Fold the bottom corner up over the filling, then fold in the left and right corners snugly over the filling like an envelope. Roll away from you, keeping it tight but not so tight that the wrapper tears. Brush the top corner with beaten egg or water and press to seal. Place seam-side down on the prepared baking sheet and repeat with remaining wrappers and filling.

Pour neutral oil into a heavy-bottomed pot, Dutch oven, or deep skillet to a depth of about 1 1/2–2 inches. Heat the oil over medium to medium-high heat to 350°F (175°C). If you don’t have a thermometer, test by dipping the tip of a wooden chopstick or spoon into the oil; small bubbles should form around it.

Working in batches, carefully lower 3–5 egg rolls into the hot oil, taking care not to crowd the pot. Fry for 3–5 minutes, turning occasionally, until golden brown and crisp on all sides. Adjust the heat as needed to keep the oil near 350°F so they brown evenly without absorbing too much oil.

Use tongs or a slotted spoon to transfer the fried egg rolls to a wire rack set over a baking sheet or to a plate lined with paper towels. Let them rest for 5–10 minutes so the filling settles and the exterior becomes even crisper.

Serve the vegetable egg rolls warm with your favorite dipping sauces such as sweet chili sauce, a soy-vinegar-sesame dip, or spicy mayo. For a party presentation, slice each egg roll on the diagonal before arranging on a platter.