Place the chicken breasts in the bottom of the slow cooker in a single layer if possible.
In a small bowl, whisk together minced garlic, minced ginger, honey, brown sugar, soy sauce, rice vinegar, and sesame oil until combined.
Pour the sauce over the chicken in the slow cooker, coating each piece.
Cook on low for 6–7 hours or on high for 3–4 hours, until the chicken is cooked through and tender.
Remove the chicken and shred it with two forks on a cutting board or in a bowl.
Strain the cooking liquid from the slow cooker into a saucepan to remove any solids, then bring the liquid to a simmer over medium heat.
Stir the cornstarch into the cold water until smooth, then whisk the slurry into the simmering sauce; continue to cook and stir until the sauce boils and thickens, about 1–2 minutes.
Return the shredded chicken to the slow cooker or a serving bowl and pour the thickened sauce over it; toss to coat evenly.
Serve topped with sesame seeds and sliced green onions, if desired.