1. Rinse chard well and trim off and discard discolored stem ends; tear leaves from stems. Thinly slice stems crosswise; cut leaves into 1-inch pieces.
2. Pour oil into a 4- to 5-quart pan over medium heat. Add chard stems and garlic; stir often until stems are limp, about 10 minutes. Add chard leaves, broth, and 3 cups water (see notes). Bring to a simmer; cook, stirring occasionally, until chard is tender to bite, 10 minutes longer.
3. Add beans and stir occasionally until hot, 1 to 2 minutes. Stir in 1/2 cup parmesan cheese and add salt and pepper to taste. Ladle into bowls; offer more parmesan to add at the table.
If you're using imported parmesan, such as Parmigiano-Reggiano, add a piece of the rind to the soup in step 2 for extra flavor.