Start with prepping all produce for this recipe. Shred cabbage as you would for coleslaw, then give it a few rough chops lengthwise so the cabbage strands aren't too long. Place into a large mixing bowl. Repeat with the shredded carrots and bean sprouts, roughly chopping so they aren't too long. Combine with the cabbage. Dice the water chestnuts and green onions and add into same bowl.
Heat a large deep skillet over medium heat with a few tablespoons of olive oil. Add a heaping teaspoon of ginger and garlic and allow them to warm in the pan for 30 seconds. Add ground chicken and cook until no more pink. Remove chicken from skillet and set aside.
Add a few more tablespoons of olive oil + 1 tablespoon of sesame oil to the skillet over medium heat. Add remaining ginger and garlic, then add the chopped veggies into the skillet. Stir and toss cabbage mixture to ensure even cooking.
Once the cabbage mixture cooks down a bit (about 5 minutes), add the teriyaki sauce and soy sauce. Add chicken (and it's juices) back into the skillet and continue to toss all ingredients until they are combined and cooked down. Add salt and pepper and taste mixture to see if any additional sauce is needed. Take off the heat and set aside to cool.
While cabbage is cooling, set up a workstation for rolling the egg rolls. I rolled on my cutting board and placed the finished rolls on a cookie sheet covered by a clean towel to prevent drying. Mix together the water and flour in a small bowl and set at workstation. Begin to heat the oil in a heavy duty pot to 375 degrees. Use a candy thermometer to monitor the heat throughout the process.
Strain out the access juices from the cabbage mixture by straining through a cheesecloth or layers of paper towels. If there is too much moisture, it will ruin the egg rolls.