Toast the bread: Preheat the oven to 400°F. Tear the Tuscan or ciabatta bread into large pieces and spread them on a large sheet pan. Drizzle with extra virgin olive oil and toss to coat evenly. Arrange the bread in a single layer and bake for about 10 minutes, until toasted and golden brown. Remove from the oven and set aside.
Cook the vegetables: Heat 3 tablespoons of extra virgin olive oil in a large pot over medium-high heat until shimmering but not smoking. Add the onions, celery, and carrots, along with a sprinkle of kosher salt. Reduce the heat to medium and cook, stirring regularly, for about 7–10 minutes, until the vegetables are softened. Add the garlic and cook for another 30 seconds to 1 minute, stirring frequently, until fragrant. Be careful not to let the garlic brown.
Add the wine: Pour in the white wine and let it reduce for a few minutes, stirring occasionally, until the liquid has reduced slightly.
Add the tomatoes and broth: Stir in the canned tomatoes, breaking them up with a wooden spoon. Once the tomatoes are broken down, add the vegetable broth. Stir to combine and bring the mixture to a simmer.
Optional: Blend the beans: If you want a smoother texture, combine about ½ cup of the tomato-broth mixture with 1 cup of the cannellini beans in a food processor and puree until smooth. (Skip this step if you prefer a chunkier soup.)
Simmer the soup: Add the Parmesan rind, thyme sprigs, bay leaf, oregano, crushed red pepper flakes (if using), and a pinch of kosher salt to the pot. Stir to combine, and let the soup simmer over medium heat for about 20 minutes to develop flavor.
Add the beans, kale, and bread: Stir in the whole cannellini beans, pureed bean mixture (if using), and chopped kale, ensuring the kale is submerged in the soup. Cook for a few minutes until the kale begins to soften. Add all of the toasted bread to the pot and stir to combine. Let the soup simmer for an additional 5–10 minutes, allowing the bread to absorb some of the liquid and soften.
Finish and serve: Remove the Parmesan rind, thyme sprigs, and bay leaf from the pot. Drizzle the soup with extra virgin olive oil, and sprinkle with grated Parmesan cheese, if desired. Serve warm and enjoy!