Preheat the oven: Set the oven to 400°F.
Roast the vegetables: Arrange the tomatoes (cut side up), onion, and garlic on a baking sheet. Drizzle with olive oil, then sprinkle with ½ teaspoon of salt and the black pepper. Roast for 45 to 50 minutes, or until the edges of the vegetables are caramelized. Let them cool for 10 minutes.
Blend the first batch: Add half of the roasted vegetables to a blender or food processor, along with 1 cup of chicken broth and 2 tablespoons of chopped basil. Blend until smooth, then transfer the mixture to a large pot.
Blend the second batch: Blend the remaining roasted vegetables with the remaining 1 cup of broth and 2 more tablespoons of basil. Add this mixture to the pot.
Simmer the soup: Heat the soup on medium. Stir in the heavy cream, Parmesan cheese, tomato paste, sugar, and the remaining 1½ teaspoons of salt. Adjust seasoning as needed, adding more cream, salt, or sugar to balance the flavor.
Serve: Ladle the soup into bowls, garnish with fresh basil and Parmesan, and pair it with grilled-cheese sandwiches for dunking.