Remove stems from chard. Chop stems and greens separately.

In a 4-qt. pot, combine broth, 2 cups water, the chard stems, beans, tomatoes, 1/2 tsp. salt, and 1/4 tsp. black pepper. Bring to a boil, then reduce heat and simmer, uncovered, 15 minutes. Stir in chard greens. Cook, uncovered, another 5 minutes, stirring occasionally.

Meanwhile, for pesto: in a small food processor, combine basil, 4 Tbsp. olive oil, 4 Tbsp. Parmesan cheese, the garlic, and 1/4 tsp. each of salt and black pepper. Process until smooth.

Brush both sides of bread slices with olive oil. On a grill or grill pan, toast bread on both sides, sprinkling with remaining cheese after turning once.

Ladle soup into bowls. Top each with pesto. Serve with bread.