Core and dice tomatoes. In a 4- to 5-quart pan, combine tomatoes, basil, olive oil, garlic, 1 teaspoon salt, and pepper to taste. Let stand until tomatoes are very juicy, about 15 minutes.
Set pan over medium-low heat and gently stir in beans. Stir occasionally until mixture is warm to touch, about 5 minutes.
Add mozzarella and parmesan cheese. Gently stir just until mozzarella has softened and starts to melt into strings, 2 to 3 minutes. Taste and stir in more salt if desired. Spoon stew into bowls and top each serving with a sprinkling of more basil and parmesan cheese.