01 - Heat olive oil in a large pot over medium heat. Add diced onion, minced garlic, and diced carrot. Cook until softened, about 5 minutes.
02 - Add crushed tomatoes, vegetable broth, sugar, dried basil, dried oregano, salt, and black pepper. Bring mixture to a boil, then lower heat and simmer for 20 minutes, stirring occasionally.
03 - Remove pot from heat and blend soup until smooth using an immersion blender or a countertop blender in batches.
04 - Stir in heavy cream and unsalted butter. Return soup to low heat and warm through, adjusting seasoning as needed.
05 - Butter one side of each bread slice. Assemble sandwiches by placing cheese slices between two bread slices, buttered sides facing outward.
06 - Heat a skillet over medium heat and cook sandwiches until golden brown and cheese is melted, approximately 2 to 3 minutes per side.
07 - Remove sandwiches from skillet and let cool for 1 minute. Cut each sandwich into strips to create dippers.
08 - Ladle the hot soup into bowls and serve alongside the grilled cheese dippers.