1.Soak beans and kelp in cold water overnight; drain. Discard kelp.
2.Place beans, garlic and 2 of the bay leaves in large saucepan with enough water to cover; bring to the boil. Reduce heat; simmer 2½ hours or until tender. Drain. Rinse under cold water; drain.
3.Preheat oven to 180°C/350°F.
4.Heat half the oil and half the butter with remaining bay leaf in large saucepan; cook onion, stirring, 15 minutes. Remove from pan.
5.Add a little more oil to pan; cook carrot, celery and daikon, stirring occasionally, 15 minutes or until golden. Remove from pan.
6.Add remaining oil; cook eggplant and burdock, 5 minutes. Remove from pan.
7.Melt remaining butter in pan; cook flour, stirring, 1 minute. Whisk in mirin and sake until smooth. Stir in stock, tomato, miso, sauce, thyme, ginger and rind. Bring to the boil, then simmer until slightly thickened.
8.Place beans, vegetables and sauce in 5-litre (20-cup) ovenproof dish; bake, covered, 2 hours. Sprinkle with breadcrumbs and parsley; bake, uncovered, further 30 minutes or until golden.