Heat the butter and olive oil together in a large pot or Dutch oven over medium heat. Add the diced onion and carrot, cooking for 5-6 minutes until softened and the onion is translucent. Add the garlic and cook for 30 seconds until fragrant — don't let it brown. The combination of butter and olive oil gives you both richness and a higher smoke point.

Add the crushed tomatoes with all their juice, chicken broth, sugar, dried oregano, salt, and pepper. Stir everything together and bring to a boil. Then reduce the heat to low and let it simmer for 15 minutes. The sugar balances the natural acidity of the tomatoes.

Remove the pot from heat and let it cool for a few minutes. Use an immersion blender to puree the soup until completely smooth and velvety. If using a regular blender, work in small batches and leave the lid slightly cracked to let steam escape. For the absolute smoothest result, a regular blender produces a silkier texture than an immersion blender.

Return the soup to low heat. Stir in the heavy cream and freshly grated Parmesan cheese. Keep stirring until the cheese is fully melted and the soup is creamy and glossy. If you're using balsamic vinegar, add it now — it adds a subtle tang that elevates the flavor.

Stir in the chopped fresh basil. Adding it at the end preserves its bright green color and aromatic flavor. Taste the soup and adjust the seasoning — you may need a little more salt or pepper.

Ladle the soup into warm bowls. Top with extra grated Parmesan, a few fresh basil leaves, a decorative swirl of cream, crunchy croutons, and cracked black pepper. Serve immediately with your favorite grilled cheese sandwich.