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