Spread carrots in a 6-quart slow cooker. Season chicken with salt and pepper.
Whisk both mustards, honey, broth, vinegar, and garlic powder. Reserve 1/3 cup in refrigerator.
Set chicken over carrots; pour remaining sauce around it. Cover.
Cook on low 3 hours; add green beans around chicken.
Cover; cook 30 to 60 minutes more until breasts reach 165°F and beans are crisp-tender.
Transfer chicken and vegetables to a platter; stir reserved sauce into cooker.
Mix cornstarch with cold water; add to cooker. Cook on high about 6 minutes until glossy; spoon over chicken.