Bring a large pot of salted water to a boil. Add the pasta and follow the package instructions to cook al dente.
As the pasta cooks, dip one of the kale leaves into the boiling water several times until it has lost its toughness, then set aside. Once the pasta is done, reserve 1⅓ cup of pasta water, strain and set the pasta aside.
Remove the veins from the remaining kale leaves and roughly chop or tear them.
In a blender, blend one can of beans, the leafy portion of the steamed kale leaf, ⅓ cup pasta water, and ½ tsp salt until smooth, then set aside.
To a large deep skillet, add the uncased sausage. Cook over medium heat, breaking it apart as it cooks. Once the sausage is fully cooked, add the butter, garlic, and cook until the garlic is golden brown, about 3 minutes.
Add the sweet vermouth and cook for 3 minutes or until the smell of alcohol has disappeared. Next, add the chicken broth and torn kale leaves. Bring to a high simmer and let cook for 3 minutes, stirring occasionally.
Add the remaining beans, pasta, salt, reserved pasta water, blended beans and greens, and grated Parmigiano Reggiano. Stir, taste, adjust for salt, and serve warm.