Cut the blanched tomatoes in half and then half again; set aside.
In a large Dutch oven over medium heat, add oil and when hot, add onions; sauté until translucent, about 2 minutes. Add garlic and sauté for only 1 minute. Add the blanched tomatoes, sugar and chicken broth. Stir well and bring to a boil and reduce heat to medium. Add basil, cheese, thyme, oregano, more salt and freshly ground black pepper. Stir and gently simmer for 20 minutes, stirring occasionally and breaking the tomatoes into smaller pieces with the tip of a spoon.
When time is up, add balsamic vinegar and process until mostly smooth using an immersion blender (or blender). Add heavy cream and stir; taste and adjust if needed. Process again with blender until the soup is perfectly smooth. Makes 6 servings.