01 - Heat the olive oil in a large pot over medium heat. Add the chopped onion, minced garlic, and sliced carrot. Sauté for approximately 5 minutes, stirring occasionally, until the onion is translucent and the vegetables have softened.
02 - Add the chopped tomatoes to the pot and cook for another 5 minutes, stirring frequently, until the tomatoes begin to break down and release their juices.
03 - Pour in the vegetable broth and add the basil, salt, pepper, and sugar if using. Bring the mixture to a boil, then reduce the heat to low and let it simmer uncovered for 20 minutes, allowing the flavors to meld and the vegetables to become fully tender.
04 - Remove the pot from heat. Using an immersion blender, puree the soup directly in the pot until completely smooth. Alternatively, carefully transfer the soup to a countertop blender in batches and blend until silky, then return to the pot.
05 - If using heavy cream or milk, stir it into the blended soup. Warm gently over low heat but do not bring to a boil, as this can cause the dairy to curdle.
06 - Taste the soup and adjust the salt and pepper as needed. Ladle into warm bowls and garnish with fresh basil leaves. Serve immediately alongside crusty bread or grilled cheese sandwiches.