Rinse the beans and pick over for small stones or debris. Soak the beans in cold water overnight with a 2-inch piece of kombu.
Discard the soaking liquid, and place the beans and the other ingredients for cooking them in a 6-quart pot. Add enough cold water to fill the pot half full. Bring to a boil. Simmer 1 to 1 ½ hours, until the beans are tender. Turn off the heat, add the tablespoon of kosher salt and allow to cool in the cooking liquid. Drain and discard the cooking liquid, onion, celery and kombu.
Whisk the dressing ingredients in a small bowl until emulsified.
To make the salad, transfer the cooked beans to a serving bowl. Add the celery, sliced shallot, fresh mint and (optional tuna.) Pour the dressing over the beans and toss to blend. Serve at room temperature.