In a small bowl, combine the olive oil, minced garlic, rosemary, thyme, salt, and black pepper.
Rub the herb mixture evenly over all sides of the turkey breast.
Place the halved fingerling potatoes and carrot pieces in the bottom of the slow cooker.
Set the seasoned turkey breast on top of the vegetables.
Pour the chicken broth around the sides of the turkey.
Cover and cook on low for 6 hours, until the turkey reaches an internal temperature of 165°F and the vegetables are tender.
Remove the turkey, let it rest for 5 minutes, then slice and serve with the carrots and potatoes.