Dissolve the soaking salt in 4 quarts of water in a large bowl. Add the beans and soak at room temperature for at least 8 hours, or up to 24 hours. Drain and rinse the beans well.

Adjust an oven rack to the lower-middle position and heat the oven to 250 degrees.

Heat the olive oil in a large Dutch oven over medium-high heat until shimmering. Add the onion, carrots, and celery and cook until the vegetables are softened and starting to brown, 6 to 10 minutes.

Stir in the garlic and minced mushrooms and cook until fragrant, about 1 minute.

Add the soaked beans, water, broth, and bay leaves. Bring to a simmer over high heat, using a wooden spoon to scrape up any browned bits from the bottom of the pot.

Cover the pot and transfer it to the oven. Bake until the beans are almost tender but still have a slight bite, about 1 hour to 1 hour 15 minutes.

Remove the pot from the oven and stir in the kale and drained tomatoes. Return the pot to the oven, covered, and continue to bake until the beans and greens are fully tender, 20 to 30 minutes.

Remove the pot from the oven and submerge the rosemary sprig in the soup. Cover and let the soup stand for 15 minutes to allow the herb to steep.

Discard the rosemary sprig and bay leaves. Season with salt and pepper to taste and serve with a drizzle of olive oil.