Roast the vegetables: Preheat oven to 200°C/390°F. Place tomatoes (with juices if canned), onion, red pepper (cut side down), and garlic on a baking sheet. Drizzle with olive oil and balsamic vinegar, then sprinkle with sugar and salt. Toss everything together and roast for 25–30 minutes, until soft and caramelized.
Blend the soup: Once slightly cooled, transfer the roasted vegetables to a blender (or use a heat-safe blender with the lid vented). Add tomato paste and basil, then blend until smooth. Pour into a soup pot.
Simmer: Stir in the broth and bring to a gentle simmer over low heat. Cover and cook for 10 minutes.
Finish the soup: Remove from heat, stir in the cream, and season with salt and black pepper. Keep warm while you make the grilled cheese.
Make the grilled cheese: Mix mozzarella and cheddar together. Pile about ½ cup of cheese between two slices of bread, then butter (or mayo) both outer sides generously. Cook in a skillet over medium heat until the cheese is gooey and the bread is golden brown on both sides.
Serve: Ladle the soup into bowls, drizzle with extra cream, and garnish with fresh basil. Serve hot with the grilled cheese on the side.
