Mash one can of cannellini beans: Drain 1 can of the beans; transfer beans to a medium bowl. Mash beans using a fork; set aside.

Cook bacon until crispy: Heat a large Dutch oven over medium-high. Add bacon; cook, stirring occasionally, until browned and crispy, 6 to 8 minutes. Transfer cooked bacon to a small bowl, reserving 1 tablespoon drippings in Dutch oven; discard or reserve remaining drippings for another use.

Cook vegetables in bacon fat: Add bay leaves to Dutch oven; cook, stirring constantly, over medium-high until fragrant, about 15 seconds. Stir in onion, celery, carrots, and salt; cook, stirring occasionally, until onions are lightly browned and tender, about 8 minutes. Stir in garlic; cook, stirring constantly, until fragrant, about 1 minute.

Add beans, broth, and seasonings: Add mashed beans, chicken broth, Italian seasoning, black pepper, and remaining 2 cans undrained beans to Dutch oven. Bring to a boil over high, stirring occasionally. Reduce heat to medium low; simmer until carrots are tender and soup thickens, about 10 minutes.

Add kale and bacon, then cream: Stir in kale and half of cooked bacon (about ⅓ cup); cover and cook, stirring occasionally, until kale is wilted, about 2 minutes. Stir in cream; cook, stirring constantly, until heated through, about 1 minute. Remove from heat. Season with salt to taste.

Garnish and serve: Divide soup among bowls; top each with about 2 teaspoons cooked bacon. Serve with grated Parmesan cheese and crusty bread.