Sauté onions and mushrooms over medium-high heat for about 5 minutes or until softened. Stir in soy sauce, garlic, ginger, white pepper, and apple cider vinegar.
Add half of the cabbage and carrots to the pan. Sauté them for 3 minutes, stirring constantly. Then, add the rest of it and sauté it for 3 more minutes.
Mix chili sauce and vegan mayonnaise in a bowl until they are fully combined. Drizzle it over the egg roll in a bowl along with sesame seeds and sliced green onions.
Heat a small pan to medium-high heat and add a thin layer of olive oil. Fry egg roll wrapper strips for about 30 seconds each. Enjoy them as a garnish.