1. Marinate the chicken: In a bowl large enough to hold chicken thighs, combine 2 garlic cloves, rosemary, 1/2 teaspoon thyme, and vinegar. Add chicken; toss to coat. Cover; refrigerate 1 to 2 hours.

2. Preheat oven to 375 degrees with rack in upper third. Drain beans, and place in a large stockpot with carrots, celery, onion, remaining 3 garlic cloves, bay leaf, sage, and remaining teaspoon thyme. Fill with enough cold water to cover by about 4 inches; bring to a simmer over medium-high heat. Reduce heat to medium-low; cook, stirring occasionally, until beans are tender, 1 1/2 to 2 hours.

3. Transfer bean mixture to a colander set over a bowl, reserving cooking liquid; let cool slightly. Puree half the mixture in the bowl of a food processor fitted with the metal blade. With the motor running, pour in about 1 cup reserved cooking liquid until thick and smooth. Return to pot with remaining bean mixture. Add the salt; stir to combine.

4. Brush an 8-by-2 1/2-inch baking dish with the oil. Remove chicken from marinade, and arrange in a single layer in prepared dish. Bake until chicken is cooked through, about 15 minutes.

5. Remove dish from oven; pour bean mixture over chicken, and sprinkle evenly with breadcrumbs. Set dish on a baking sheet; return to oven, and bake until breadcrumbs are golden brown and beans are bubbling, about 30 minutes. Remove from oven; serve hot.