01 - Heat olive oil in a large saucepan over medium heat. Add chopped onion and diced carrot; sauté for 5 minutes until softened and translucent.
02 - Stir in minced garlic and cook for 1 minute until fragrant, being careful not to burn.
03 - Add canned tomatoes with their juice, vegetable broth, sugar (if using), salt, and pepper. Bring the mixture to a gentle simmer.
04 - Reduce heat to low, cover, and simmer for 20 minutes, stirring occasionally to prevent sticking and allow flavors to meld.
05 - Remove saucepan from heat. Add fresh basil leaves and let them wilt slightly in the hot soup.
06 - Using an immersion blender, puree the soup until completely smooth and velvety. Alternatively, work in batches using a countertop blender, venting carefully to release steam.
07 - Stir heavy cream into the pureed soup until fully incorporated. Taste and adjust salt and pepper seasoning as needed.
08 - Reheat gently over low heat if necessary. Serve hot in bowls, garnished with additional fresh basil leaves.