Rinse beans. In a large saucepan combine beans and 4 cups cold water. Bring to boiling; reduce heat. Simmer, uncovered, for 2 minutes. Remove from heat. Cover; let stand for 1 hour. (Or, soak beans in water overnight in a covered pan.) Drain beans and rinse. In the same saucepan combine beans and 4 cups cold water. Bring to boiling; reduce heat. Cover and simmer 1 to 1-1/2 hours or until tender. Drain beans.
Preheat oven to 350°F In a large skillet, heat oil over medium-high heat. Add chicken; reduce heat to medium-low. Brown chicken about 10 minutes, turning once to brown both sides. Remove chicken from skillet. Drain all but 1 tablespoon of drippings from the skillet.
Add carrots, onion, celery, and garlic to drippings in skillet. Cover and cook about 10 minutes or vegetables are just tender, stirring occasionally. Stir in drained beans, undrained tomatoes, sausage, thyme, salt, and, if desired, cayenne pepper. Bring to boiling. Transfer mixture to a 2-quart rectangular baking dish. Arrange chicken thighs on top.
Bake, uncovered, about 25 minutes or until chicken is no longer pink (180 degrees F).
Easy version: Omit dried beans and step 1. Add 2 15-ounce cans rinsed and drained Great Northern beans to the vegetable mixture in step 3 with the tomatoes.