Heat 1 tablespoon oil in a large skillet over medium-high heat. Saute chicken and ginger in hot oil until chicken is cooked through, 7 to 10 minutes. Set aside.
Whisk vinegar, teriyaki sauce, remaining 1 tablespoon oil, and honey together in a bowl. Add chicken mixture, cherries, carrots, green onion, and almonds; toss to combine.
Spoon 1/12 the chicken/cherry mixture onto the center of each lettuce leaf; roll leaf around filling and serve.
