Arrange the boneless chicken breasts evenly at the base of a slow cooker insert.
In a separate mixing bowl, combine the freshly minced garlic, grated ginger, soy sauce, toasted sesame oil, rice vinegar, honey, and brown sugar. Whisk thoroughly until all ingredients are blended into a smooth liquid mixture.
Gently pour the seasoned sauce mixture over the chicken, ensuring all pieces are well coated.
Secure the lid on the slow cooker. Cook on the HIGH setting for 3 hours, or select LOW and cook for 5 to 6 hours until the chicken becomes tender and easy to shred.
Carefully transfer the fully cooked chicken breasts to a cutting board or plate. Using two forks, pull apart the meat until finely shredded.
Strain the remaining teriyaki liquid from the slow cooker into a medium saucepan to remove solids. Set the saucepan over medium-high heat and bring the liquid to a simmer.
In a small bowl, mix the cornstarch with the cold water, stirring until the mixture is fully dissolved with no lumps.
Pour the cornstarch slurry into the simmering sauce. Allow the mixture to boil for 1 to 2 minutes, stirring continuously, until it thickens to a glossy, pourable glaze.
Return the shredded chicken to the slow cooker or a large serving bowl. Pour the thickened teriyaki sauce over the chicken and mix well to evenly coat the meat.
Finish by scattering sesame seeds and sliced green onions on top before serving.