Place the chicken breasts into the slow cooker.
In a separate small bowl, whisk together the minced garlic, ginger, honey, brown sugar, soy sauce, sesame oil, and rice vinegar until fully combined.
Pour the prepared sauce mixture over the chicken breasts in the slow cooker.
Cover the slow cooker and cook on HIGH for 3 to 4 hours, or on LOW for 6 to 7 hours.
Once cooked, remove the chicken breasts from the slow cooker and shred them using two forks.
Strain the teriyaki sauce from the slow cooker into a saucepan, discarding any solids.
Set the saucepan over medium-high heat and bring the sauce to a simmer.
In a small bowl, dissolve the cornstarch in the cold water, then pour the mixture into the simmering sauce.
Allow the sauce to boil for 1 to 2 minutes, or until it thickens slightly.
Drizzle the thickened sauce over the shredded chicken and toss to coat.
Garnish the chicken with sesame seeds and sliced green onions, then serve immediately.