Preheat the oven and place tomatoes in a roasting tray with olive oil, balsamic vinegar, sugar, and salt. Roast until soft and slightly caramelized. This step concentrates flavor and removes raw acidity.
Heat olive oil in a large pot and cook onions slowly until soft and translucent. Add garlic and basil, then stir in roasted tomatoes, tomato paste, and broth. Let everything simmer gently.
Once the vegetables are tender, blend the soup until smooth and creamy. Use a blender or immersion blender. Return to low heat and stir in cream, salt, and pepper.
Mix mozzarella and cheddar together. Fill buttered bread slices with cheese and grill on medium heat. Cook slowly so the cheese melts before the bread browns.
Pour hot soup into bowls and serve with grilled cheese on the side or sliced for dipping. The contrast of creamy soup and crisp sandwich is unbeatable.