01 - Heat olive oil in a large saucepan over medium heat. Add onion and carrot, cook for 5 minutes until softened.
02 - Add minced garlic and cook for 1 minute until fragrant.
03 - Add tomatoes with juice, vegetable broth, sugar, dried basil, salt, and black pepper. Bring to a boil, then reduce heat, cover and simmer for 20 minutes.
04 - Remove from heat and puree the soup until smooth using an immersion blender or countertop blender in batches.
05 - Return soup to low heat, stir in heavy cream, and heat gently for 2 to 3 minutes. Adjust seasoning if needed.
06 - Butter one side of each slice of sandwich bread.
07 - Place four buttered slices, buttered side down, on a skillet over medium heat. Top each with two slices of cheddar cheese, then cover 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 slice each into strips for dipping.
10 - Ladle hot soup into bowls and serve with grilled cheese dippers alongside.