Place large skillet on stovetop over medium heat. When pan is hot, add 1 tablespoon sesame oil. Continue heating pan until oil is hot and shimmery, swirling and tilting pan as needed to distribute oil across surface.
When oil is hot, place pieces of 1 pound boneless, skinless chicken breasts in skillet. Cook chicken, stirring occasionally, approximately 3 to 5 minutes or until chicken pieces are opaque and lightly golden.
Transfer cooked chicken to plate or bowl and set aside.
Return skillet to heat. Add remaining 1 tablespoon sesame oil to skillet and heat oil until shimmery.
When oil is hot, add sliced white parts of 6 green onions, ½ cup diced red onion, and 2 ½ tablespoons minced fresh garlic to skillet. Stir to incorporate, then sauté ingredients, stirring frequently, approximately 5 minutes or until onions begin to soften.
Once onions have softened, add 1 teaspoon grated fresh ginger, 1 8-ounce can water chestnuts, 1 tablespoon sriracha, 1 14-ounce bag coleslaw mix, 3 tablespoons coconut aminos, 1 tablespoon rice wine vinegar, ¼ teaspoon freshly cracked white pepper, and salt to skillet. Stir ingredients together until well incorporated.
Cook egg roll mixture, stirring often, until cabbage is tender, approximately 5 minutes.
Reduce heat under skillet to low. Return cooked chicken to skillet and stir to incorporate.
Continue cooking mixture, stirring often, until chicken is warmed through and liquid has reduced.