First, make the gingered carrots and broccoli. Heat a large pan over medium heat, add just enough oil to cover the bottom of the pan, add 2 tablespoons of ginger, 2 cloves of garlic, and let cook for 1 minute before adding the carrots and broccoli stems. Cook for 7 minutes, then set aside.
Next, make the cabbage. Return the pan to the heat, coat the bottom with a thin layer of oil once again, add the remaining tablespoon of ginger, cook for 1 minute. Then, add the cabbage and cook for 7 minutes, stirring occasionally, and adding 1- 2 tablespoons of water about halfway through to help the cabbage cook. When done, remove from heat.
Meanwhile, you can make the tahini dressing. Add the tahini and water to a small bowl and mix with a fork until smooth. Then add the lemon juice, and mix well again. Finally mix in salt to taste. Set aside.
Once the veggies are all cooked, you can assemble the wraps. Place a large shallow dish or rimmed baking sheet in your sink, then fill with hot water from the faucet. Working one at a time, submerge one spring roll sheet in the hot water for 3-5 seconds, then quickly remove to a clean cutting board and lie flat. Place a few slices of avocado in the middle, and on top of that about ¾ cup of mixed veggies - including the cabbage, carrots, and broccoli.
Then fold the top and bottom in towards the center, then the left side, and then the right to create a spring roll or pocket type shape. Repeat with remaining ingredients, working one at a time.
Serve the ginger veggie wraps with the tahini dipping sauce. These save well and can be eaten cold.