Heat the oil in a medium pot over medium heat. Add the sausage and cook, breaking the meat up with a wooden spoon, until browned, about 4 minutes. Add the onions and celery and continue to cook, stirring, until the vegetables are softened, about 5 minutes. Add the beans, tomatoes, broth and 4 cups water and increase the heat to medium-high to bring to a simmer. Continue to simmer, adjusting the heat if necessary, until the vegetables are tender, about 10 minutes.
Add the tortellini and continue to cook until they are tender, about 10 minutes. Season to taste with salt and pepper. Serve in bowls topped with a spoonful of pesto and a sprinkle of grated Parmesan.