Place chicken in a large bowl. Whisk soy sauce, ginger, tarragon, and brown sugar together in a bowl; pour over chicken. Cover bowl with plastic wrap and marinate in refrigerator, 2 to 4 hours.
Remove chicken from marinade with a slotted spoon, reserving marinade.
Heat vegetable oil in a skillet over medium-high heat; saute chicken until no longer pink in the center, about 5 minutes. Add marinade; bring to a boil for 3 minutes. Stir cashews and carrots into chicken.
Arrange cabbage and baby kale on 4 plates. Drizzle sesame oil over each cabbage-kale serving; top with chicken mixture.