Heat the oil in a heavy 5- to 6-qt. Dutch oven over medium heat. Add the onion and cook, stirring occasionally, until tender, 5 to 6 minutes. Add the sausage, raise the heat to medium high, and cook, stirring and breaking up the sausage with a wooden spoon or spatula until lightly browned and broken into small (1-inch) pieces, 5 to 6 minutes. Add the garlic and mushroom then cook for 1 minute, now stir in the beans. Add the kale to the pot in batches; using tongs, toss with the sausage mixture to wilt the kale and make room for more.