Dissolve the salt in cold water in a large container. Add the beans and soak at room temperature for at least 8 hours or up to 24 hours — this brine helps the skins remain intact during the long cook. Drain and rinse the beans thoroughly.

Adjust the oven rack to the lower-middle position and heat the oven to 250°F. Heat the oil in a Dutch oven over medium heat until shimmering. Add the onion, carrot, chard stems, and porcini mushrooms and cook until the vegetables have softened, about 7 to 10 minutes.

Stir in the white mushrooms and cover the pot. Cook until the mushrooms have released their liquid, about 5 minutes. Uncover the pot and continue to cook until the liquid has evaporated and the mushrooms are dry and deeply browned, about 5 to 10 minutes.

Stir in the garlic and thyme and cook until fragrant. Add the broth, soaked beans, and bay leaves and bring the liquid to a boil. Cover the pot, transfer it to the oven, and cook until the beans are almost tender, about 1 to 1¼ hours.

Stir in the chard leaves and tomatoes. Return the pot to the oven and continue to cook, covered, until the beans and vegetables are fully tender, 30 to 40 minutes. Remove from the oven, discard the bay leaves, and season with salt and pepper to taste before serving hot.