Heat the sesame oil in a medium skillet over medium-high heat. Add the green onions, garlic, and ginger and cook for 1 minute. Add the mushrooms and cook until soft and add the cabbage, carrots, and crushed red pepper. Cook until all veggies are soft and add the soy sauce and vinegar. Turn off heat.
Lay the egg roll wrappers on a work surface and spoon about a tablespoon of the vegetable filling in each wrapper. Brush the edges with the egg and fold in half, pressing to seal. Lay on a parchment-lined sheet pan. Brush with egg and bake for 10-15 minutes until lightly browned and crisp to the touch.
While the wontons are baking, make the honey sesame sauce by combining the ingredients in a jar with a lid. Shake until well combined.
Serve the wontons with the sauce, and extra green onions if desired.