Heat the oil in a large skillet or wok over medium-high heat. Add the cabbage, carrot and mushrooms, and cook for 5 to 6 minutes, stirring often, until the cabbage wilts and the mushrooms release their liquid.
Stir in the bean sprouts, scallions, garlic and ginger. Cook 1 minute more, then add the soy sauce and sesame oil.
Sprinkle the cornstarch over the vegetables and stir well; this soaks up extra liquid so the filling holds together instead of running. Remove from the heat and cool 10 minutes.
Lay a wrapper in a diamond shape and spoon 1/4 cup of the cooled filling onto the lower third.
Fold the bottom point over the filling, fold in both sides, and roll tightly, sealing the final edge with water (no egg needed to keep this vegan).
Repeat with the remaining wrappers, keeping unused ones under a damp paper towel.
Air fry at 390°F: arrange seam side down with space between rolls, spray with oil, and cook 7 minutes. Flip and cook 3 minutes more, until golden and crisp at the seams.
Or bake at 425°F on a parchment-lined sheet for 14 to 16 minutes, flipping once, until crackling and browned.
Whisk the soy sauce, rice vinegar and ginger together for the dip, and scatter the sesame seeds over the top just before serving.
Rest the egg rolls 2 minutes before serving so the filling settles and does not spill on the first bite.
Good to Know
Chop the mushrooms fine and squeeze them in a paper towel after cooking if they look watery; dry filling means a crisp wrapper.