Roast the Tomatoes to Enhance Flavor: Preheat your oven to 200°C (390°F). Place the tomatoes in a roasting pan or baking dish. Drizzle with balsamic vinegar and olive oil, then sprinkle with sugar and salt. Toss gently to coat all tomatoes evenly. Roast in the oven for 25–30 minutes until the tomatoes soften, break apart, and develop a rich caramelized color.
Sauté the Aromatics: While the tomatoes roast, heat a splash of olive oil in a large heavy-bottomed pot or Dutch oven over medium heat. Add the finely chopped onions and cook, stirring occasionally, until they turn translucent and fragrant, about 5–7 minutes. Add minced garlic and whole basil leaves; cook for another minute to release their aromas.
Combine and Simmer the Soup Base: Add the roasted tomatoes (including any juices) to the pot along with the tomato paste and a pinch of sugar to balance acidity. Stir well to combine all ingredients. Pour in the broth, bring the mixture to a gentle boil, then reduce heat to low. Cover and let simmer for 10 minutes, allowing flavors to meld.
Puree the Soup to Smooth Perfection: Remove the pot from heat. Using an immersion blender, carefully puree the soup until silky smooth. Alternatively, transfer the soup in batches to a countertop blender, then return to the pot.
Finish with Cream and Seasoning: Stir in the heavy cream for a luscious, velvety texture. Taste and adjust seasoning with salt and pepper as needed.
Prepare the Grilled Cheese Sandwiches: In a bowl, mix together shredded mozzarella and cheddar. Generously butter one side of each bread slice. Place half of the slices, butter side down, on a clean surface and evenly spread about ½ cup cheese mixture on each. Top with remaining bread slices, butter side up, to form sandwiches.
Grill Sandwiches to Golden Brown: Heat a non-stick skillet over medium heat. Place sandwiches in the pan and cook until the bread is golden brown and crisp, about 3–4 minutes per side, while the cheese melts completely inside.
Serve and Garnish: Ladle the warm tomato soup into bowls. Add a swirl of cream and a few fresh basil leaves on top for garnish. Serve immediately alongside the hot grilled cheese sandwiches for a perfect, comforting meal.