Heat the olive oil or butter in a large pot over medium heat. Add the diced onion and cook for 5–7 minutes, stirring occasionally, until softened but not browned.

Add the minced garlic and cook for 30–60 seconds, stirring constantly, until fragrant. Do not let the garlic become deeply browned, as it can turn bitter.

Add the chopped tomatoes, broth, kosher salt, black pepper, and optional sugar to the pot. Bring the mixture to a gentle boil.

Reduce the heat, partially cover the pot, and simmer for 25–30 minutes. Continue until the tomatoes are very soft and the broth has taken on a rich tomato flavor.

Stir in the fresh basil leaves. Blend the soup until smooth with an immersion blender, or carefully blend it in batches in a countertop blender with the lid vented to release steam.

Return the blended soup to the pot and stir in the heavy cream, if using. Warm gently without boiling, then taste and adjust the salt, pepper, and acidity.

Ladle the soup into bowls and finish with fresh basil, Parmesan, olive oil, croutons, or toasted crusty bread. Serve hot.