Place the chicken, carrots, and sweet potatoes in a 2- or 3-quart slow cooker.
In a small bowl, whisk together the honey, balsamic, tomato paste, black pepper, salt, garlic, ginger, and red pepper flakes. Pour the glaze over the chicken.
Cover and cook on low for 6 to 8 hours (or, 3 to 4 hours on high).
Add the broccoli for the last 30 minutes of cooking.
Serve the chicken and vegetables topped with 1/2 cup of the honey glaze.