Heat oil in a large pot over medium-high heat. Cook sausage until browned, stirring often, 10 minutes. Add broth, artichokes, and 2 cups water and bring to a boil.
Reduce heat to a simmer. Add chard and cook, covered, until wilted, about 3 minutes. Ladle soup into bowls and serve with freshly grated cheese and bread.
Note: Nutritional analysis is per 1 1/2-cup serving.