Step 1Heat 1 tablespoon of oil in a 3-quart pot over medium-high heat. Add the garlic and onion. Cook, stirring often, until the onion is softened, about 5 minutes.
Step 2Reserve 3/4 cup of the beans and place the rest along with the rosemary in the bowl of a food processor and puree until smooth.
Step 3Transfer the puree and the reserved beans to the pot. Add the vegetable broth, vinegar, tomatoes, mushrooms, salt, 1/4 teaspoon red pepper flakes (or more to taste) and pepper to taste. Bring to a boil, then reduce the heat and simmer, covered, stirring occasionally, about 20 minutes. (The chowder can be made 2 days ahead to this point and refrigerated, or frozen up to 1 month. To serve, gently reheat the soup until hot.)
Step 4Stir in the spinach leaves. Cook 1 minute. Taste; adjust the seasoning.
Step 5Serve hot, drizzled with the remaining 1 tablespoon of oil and garnished with parsley or chives, if using. Pass the Parmesan cheese.