Heat the oil in a Dutch oven over medium heat. Add the sausage and cook, without stirring, until browned, about 3 minutes. Turn, add the garlic, and cook for 2 minutes more. Add the beans, broth, and tomatoes and their juices. Increase heat to medium-high and bring to a simmer.
Fold each kale leaf in half lengthwise and remove the stem by ripping or cutting it. Tear each leaf into large pieces. Add the kale to the pan and cook, stirring occasionally, until wilted, 2 to 3 minutes. Remove from heat. Season with the salt and pepper and spoon into individual bowls. Serve with the bread (if using).