Place the chicken breasts in the slow cooker.
Whisk the garlic, ginger, honey, brown sugar, soy sauce, sesame oil, and rice vinegar. Pour the mixture over the chicken. Cover and cook on HIGH for 3 hours or LOW for 5–6 hours, until the chicken reaches 165°F (74°C).
Remove and shred the chicken. Strain the cooking sauce into a saucepan and bring it to a simmer. Mix the cornstarch with the cold water, whisk it into the sauce, and boil for 1–2 minutes until just thickened.
Pour the sauce over the shredded chicken and toss to coat. Garnish with sesame seeds and green onions. Serve warm.