Place the chicken breasts in a 5-6 quart slow cooker.
In a small bowl, whisk together the garlic, ginger, honey, brown sugar, soy sauce and rice vinegar. Pour the soy sauce mixture over the chicken. Cover and cook on HIGH for 3-3 1/2 hours or on LOW for 5-6 hours. (high is recommended).
Transfer the chicken to a cutting board and shred into large chunks with two forks.
Pour the sauce from the slow cooker through a strainer and into a medium saucepan.
In a small bowl, whisk together the water and cornstarch until dissolved. Whisk into the saucepan and bring to a simmer over medium heat. Cook for 1-2 minutes or just until sauce has thickened.
Place the chicken back into the slow cooker, pour in the sauce and toss to coat.
Serve with rice and garnish with sesame seeds and sliced green onions.
Notes
Alternatively, you can wish the slurry into the sauce with the chicken. It will thicken either way. Straning just creates a smoother sauce.