1. Combine the broth, diced tomatoes, tomato sauce, onion, garlic, salt, pepper, and 1/4 cup of the basil in a large Dutch oven over medium-low. Cook, stirring occasionally, until the onion and garlic are softened, about 30 minutes. Puree the soup with an immersion blender until smooth.

2. Melt 3 tablespoons of the butter in a medium skillet over medium. Whisk in the flour, and cook, whisking constantly, until the mixture is lightly browned, about 2 minutes.

3. Gradually whisk in the heavy cream, whisking until the mixture is thickened and smooth. Gradually stir the cream mixture and Parmesan into the tomato mixture; cook, stirring occasionally, over medium-low until the cheese melts and is well incorporated, about 15 minutes.

4. Meanwhile, melt the remaining 2 tablespoons butter in a medium skillet until foamy. Place 2 bread slices in the skillet; top each bread slice with American cheese slices, 2 tablespoons of the chopped basil, and 1 bread slice.

5. Cook until the bread is golden brown and the cheese melts, about 3 minutes per side. Cut each sandwich into 9 pieces. Serve the soup with the croutons, and sprinkle with the remaining 1/4 cup basil.