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