01 - Heat olive oil in a large pot over medium heat. Add onion, garlic, and carrot; sauté for 5 minutes until softened and fragrant.
02 - Add chopped tomatoes to the pot and cook for another 5 minutes, stirring occasionally until they begin to break down.
03 - Pour in vegetable broth, then add basil, salt, pepper, and sugar if using. Bring to a boil, then reduce heat and simmer uncovered for 20 minutes until all vegetables are tender.
04 - Remove from heat. Use an immersion blender directly in the pot, or transfer to a countertop blender in batches, and blend until silky smooth.
05 - If using, stir in heavy cream or milk. Warm through gently but do not boil.
06 - Taste and adjust salt and pepper as needed. Ladle into bowls and garnish with fresh basil leaves.