01 - Heat olive oil in a large saucepan over medium heat. Add chopped onion and diced carrot, sauté for 5 minutes until softened.
02 - Stir in minced garlic and cook for 1 minute until fragrant.
03 - Add whole peeled tomatoes with their juice, vegetable broth, sugar, dried basil, salt, and pepper. Bring to a boil, then reduce heat, cover, and simmer for 20 minutes.
04 - Remove from heat and puree the mixture until smooth using an immersion blender or a countertop blender in batches.
05 - Return soup to low heat, stir in heavy cream, and heat gently for 2 to 3 minutes. Adjust seasoning to taste.
06 - While soup simmers, butter one side of each bread slice evenly.
07 - Place four bread slices, buttered side down, on a skillet preheated to medium heat. Layer two slices of cheddar cheese on each, then top with remaining bread slices, buttered side up.
08 - Cook sandwiches for 3 to 4 minutes per side until golden brown and cheese is melted.
09 - Remove sandwiches from skillet, let cool slightly, then cut each into strips for dipping.
10 - Ladle hot soup into bowls and serve immediately alongside the grilled cheese dippers.