01 - Heat olive oil in a large saucepan over medium heat. Add chopped onion and carrot; sauté for 5 minutes until softened and fragrant.
02 - Stir in minced garlic and cook for 1 minute until aromatic, being careful not to burn.
03 - Add tomatoes with their juices, vegetable broth, sugar, salt, pepper, and oregano. Bring to a boil, then reduce heat and simmer uncovered for 20 minutes to develop flavors.
04 - Remove from heat and stir in chopped fresh basil leaves, allowing them to wilt slightly in the hot soup.
05 - Use an immersion blender to puree the soup until completely smooth and velvety. Alternatively, transfer in batches to a countertop blender, covering the lid with a towel to prevent splattering.
06 - Return the soup to low heat. Stir in heavy cream and heat gently for 2-3 minutes until warmed through; do not let the soup boil to prevent curdling.
07 - Taste and adjust salt and pepper as needed. Ladle into warmed bowls and garnish with additional fresh basil leaves and a drizzle of olive oil or cream.