Finely chop enough rosemary leaves to get 1 tablespoon. Set the remaining sprigs aside. In a small bowl, combine the chopped rosemary, 1 ½ teaspoons of the salt, the paprika, garlic powder, onion powder, dried thyme, and pepper.
Place the potatoes in one side of a 6-quart oval slow cooker and put the carrots in the other side. Season with the remaining ½ teaspoon salt. Top with 2 rosemary sprigs.
Rub the spice mixture all over the chicken and stuff the cavity with the remaining 4 rosemary sprigs. Transfer the chicken to the slow cooker, putting it on top of the vegetables, breast side up.
Cover and cook on low for 8 hours, until the chicken is cooked through and falls off the bone and the vegetables are tender. Using a wide spatula, carefully transfer the chicken to a dish or platter. Transfer the vegetables to a separate platter. Discard the skin and rosemary sprigs and carve the chicken. Place the jus in a gravy separator and serve with the chicken and vegetables.
