01 - Heat olive oil in a large saucepan over medium heat. Add onion and 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 black pepper. Bring to a boil, reduce heat to low, cover, and simmer for 20 minutes.
04 - Remove from heat and puree the mixture until smooth using an immersion blender or countertop blender in batches.
05 - Return pureed soup to low heat, stir in heavy cream, and heat gently for 2 to 3 minutes. Adjust seasoning to taste.
06 - Butter one side of each bread slice evenly.
07 - Place four bread slices, buttered side down, onto a skillet over medium heat. Top each slice with two cheddar cheese slices, then cover with remaining bread slices, buttered side up.
08 - Cook for 3 to 4 minutes per side until the bread is golden brown and cheese melts.
09 - Remove sandwiches from skillet, allow to cool slightly, then cut each into strips suitable for dipping.
10 - Ladle hot tomato soup into bowls and serve with grilled cheese dippers alongside.