In a medium bowl, whisk together honey, soy sauce, minced garlic, minced ginger, and sesame oil until well combined.
Place the chicken breasts in the bottom of the slow cooker. Season them with salt and pepper.
Pour the honey garlic mixture over the chicken breasts, ensuring they are well coated.
Add the diced onion, sliced bell pepper, broccoli florets, and sliced carrots to the slow cooker. Stir gently to distribute the vegetables evenly around the chicken.
Cover the slow cooker and cook on low for 6-7 hours or on high for 3-4 hours, until the chicken is cooked through and vegetables are tender.
If you prefer a thicker sauce, mix cornstarch with a tablespoon of cold water to create a slurry. Stir this into the slow cooker about 30 minutes before serving and increase the setting to high.
Once cooked, shred the chicken using two forks and mix everything together in the sauce.
Serve the honey garlic chicken and veggies over rice or quinoa, garnished with sliced green onions and a sprinkle of sesame seeds.