In a medium bowl, whisk together the honey, soy sauce, minced garlic, grated ginger, sesame oil, and rice vinegar until well combined.
Season the chicken breasts with salt and pepper on both sides. Place them in the slow cooker.
Pour the honey garlic mixture over the chicken, ensuring the pieces are well coated.
Add the sliced carrots around the chicken, and tuck the broccoli florets on top.
Cover the slow cooker with the lid and cook on low for 4-6 hours, or until the chicken is cooked through and tender. If your slow cooker has a high setting, cook for 2-3 hours.
Once cooked, remove the chicken and vegetables from the slow cooker and place them on a serving dish.
If desired, you can thicken the sauce by pouring it into a pan over medium heat for a few minutes, allowing it to reduce.
Drizzle the thickened sauce over the chicken and vegetables. Garnish with chopped green onions and sesame seeds before serving.
Notes
For a gluten-free option, use tamari instead of soy sauce.