Heat 1 1/2 teaspoons oil in a Dutch oven over medium heat. Add carrots and next 4 ingredients (through garlic); cook 5 minutes or until onion is tender, stirring frequently. Add chard and next 8 ingredients (through tomatoes); stir to combine. Bring to a boil; reduce heat, and simmer 45 minutes.
Place 1/3 cup beans in a bowl; mash with a fork until smooth. Add mashed beans, remaining beans, and bread to soup; cook 5 minutes or until thoroughly heated. Ladle soup into bowls. Drizzle remaining 2 tablespoons oil evenly over soup.