Spray or line (with a slow cooker bag) a 5-6 quart slow cooker.
In a heavy large skillet, heat the 1 tbs of oil and brown chicken very well on both sides. Place browned chicken in slow cooker.
Saute garlic and ginger in the same pan you browned the chicken for about 1 minute. Add the 1/4 cup water, scraping up all the browned bits at the bottom of the pan. Add the soy sauce, honey, sherry, molasses and vinegar. Stir well and pour over chicken in slow cooker.
Cover slow cooker and cook on HI for 4-5 hours (or until chicken is done).
Remove the chicken from the slow cooker. Add the sesame oil to the remaining slow cooker juices. Mix cornstarch with 1/4 cup water. With slow cooker set to HI, slowly add the cornstarch mixture and stir until sauce is thickened. Let bubble about 1 minute to swell and cook the cornstarch.
NOTE: You may want to remove the juice to another pan if you used skin-on chicken as there will be fat. It will be easier to skim the fat if you remove to another pan.
Preheat broiler to high. Place chicken on a broiler pan. Broil about 3 minutes each side or until chicken surface is slightly caramelized and bubbly.
Place chicken on serving tray and pour sauce over chicken. Garnish with sliced green onions.