Roast the tomatoes -- Heat the oven to 250°F. Line a baking sheet with parchment or foil. Core the tomatoes and cut them in half lengthwise. Put them on the baking sheet, cut side up, and sprinkle with the salt. Bake the tomatoes until they look dry but are still slightly plump and not leathery, 4 to 6 hours, depending on their size. Cut any large pieces in half.
Cook the beans -- Drain the soaking beans and put them in a large pot along with the carrot, onion, and bay leaves. Tie the parsley and thyme together and add them to the pot. Add enough water to cover by 2 inches. Bring to a boil and then reduce to a simmer. Cover and cook until tender, about 2 hours. Check that the beans stay covered with liquid, adding more if needed. When the beans are tender, add the salt. (The beans can be cooked up to 2 days ahead. Remove the carrot, onion, herbs, and bay leaves and refrigerate the beans in their liquid.)
In a Dutch oven or high-sided skillet, heat the olive oil over medium. When the oil is hot, add the garlic and cook for about 1 minute. Add the celery and cook until softened slightly, about 2 minutes. Drain the beans, reserving their cooking liquid. Add the beans and 2 cups of the cooking liquid to the celery and garlic. Add the slow-roasted tomatoes and season with salt and pepper. Bring to a simmer and cook for 2 to 3 minutes. (At this point, the dish can be covered and held up to an hour at room temperature.)