In a 1-1/2- or 2-quart slow cooker combine carrots, celery, corn, onion, garlic, rosemary, and the 1/4 teaspoon pepper. Top with chicken. Pour broth over all in cooker.
Cover and cook on low-heat setting for 7 to 8 hours or on high-heat setting for 3-1/2 to 4 hours. If no heat setting is available, cook for 5 to 5-1/2 hours.
If using low-heat setting, turn to high-heat setting (if no heat setting is available, continue cooking). Remove chicken from cooker. Transfer chicken to a cutting board; cool slightly. When cool enough to handle, remove chicken from bones; discard bones. Chop chicken; return to mixture in cooker. In a small bowl combine milk and flour until smooth. Stir into mixture in cooker.
Prepare Cornmeal Dumplings. Using two spoons, drop dough into four mounds on top of hot chicken mixture. Cover and cook for 20 to 25 minutes more or until a toothpick inserted into a dumpling comes out clean. (Do not lift cover during cooking.) If desired, sprinkle each serving with coarse pepper.
In a medium bowl combine flour, cheese, cornmeal, baking powder, and salt. In a small bowl combine egg, milk, and melted butter. Add egg mixture to flour mixture. Stir with a fork until moistened.