Heat 1 tablespoon of the oil in a heavy pot or heavy large saucepan over medium-low heat. Add the sausages and cook until brown on all sides, about 10 minutes. Transfer the sausages to a plate. Add the remaining 2 tablespoons oil, the carrots, onion, rosemary, and pepper flakes to the pot and sauté until the vegetables start to soften, about 5 minutes. Add the mushrooms and sauté until they begin to soften, about 5 minutes.
Add 1 quart of broth, beans with their liquid, tomatoes, and Parmesan rind to the pot. Slice the sausages into ¼- to ½-inch thick rounds and add to the pot with any juices on the plate. Bring the soup to a boil, cover partially, reduce the heat and simmer until the broth is rich and flavorful, about 30 minutes.
Season the soup to taste with salt and pepper. Add the pasta and simmer uncovered until beginning to soften, about 5 minutes. Add the kale and simmer until the kale and pasta are tender, about 5 minutes longer, thinning with more broth if too thick. Serve, grating Parmesan cheese over, if desired.