Prepare the Filling:

1. Cook the Veggies: Heat the sesame oil in a large skillet over medium heat. Add the garlic and fresh ginger, sautéing for about 1-2 minutes until fragrant. Add the shredded cabbage, carrots, and mushrooms (if using). Stir and cook for about 5-7 minutes, until the vegetables soften.
2. Season the Filling: Add the soy sauce, rice vinegar, and sugar. Stir to combine and cook for another 2 minutes. Taste and adjust seasoning with salt and pepper. Remove from heat and let the mixture cool slightly.
3. Add the Sesame Seeds and Green Onions: Stir in the sesame seeds and green onions, mixing everything together thoroughly.

Assemble the Egg Rolls:

4. Prepare the Wrappers: Lay out your egg roll wrappers on a clean surface. Place one wrapper with a corner facing you, like a diamond shape.
5. Fill the Wrappers: Spoon a generous amount of the veggie mixture into the center of the wrapper, being careful not to overfill. Fold the bottom corner up over the filling, then fold in the sides, and roll tightly toward the top corner. Seal the edge with a little water to keep it closed. Repeat for the remaining egg rolls.

Cook the Egg Rolls:

6. Frying Method: Heat 2-3 inches of vegetable oil in a deep pan or wok over medium-high heat. Once the oil reaches 350°F (175°C), gently place the egg rolls into the hot oil, frying them in batches to avoid overcrowding. Fry for about 3-4 minutes, turning occasionally until golden brown and crispy. Remove from the oil and drain on paper towels.
7. Baking Method (Healthier Option): Preheat your oven to 425°F (220°C). Place the egg rolls on a baking sheet lined with parchment paper, brushing them lightly with sesame oil or vegetable oil. Bake for 15-20 minutes, flipping halfway through, until golden and crispy.

Prepare the Dipping Sauce:

8. Mix the Sauce: In a small bowl, combine the soy sauce, rice vinegar, sesame oil, honey, and grated ginger. Stir until well combined.

Serve and Enjoy:

9. Serve: Serve the egg rolls warm with the dipping sauce on the side.