Special equipment: an immersion blender
For the soup: In a large pot, heat the butter and oil over medium-high heat. Add the carrots, garlic and onions and stir and cook for 5 minutes. Add the tomato paste and cook, stirring, another 2 to 3 minutes. Add the canned tomatoes, broth, parsley and 1 cup water and stir. Bring to a boil, then reduce the heat, cover the pot and simmer for 15 minutes.
Use an immersion blender to puree the mixture, still leaving some texture to the tomatoes. Add the cream, basil and some salt and pepper, and simmer for another 5 minutes.
For the croutons: Drizzle the baguette rounds lightly with olive oil. Place little piles of Parmesan in a nonstick skillet. Place a baguette round on top of each pile. Turn on the heat to medium. Press with a spatula as you melt the cheese into each round. When set, flip to the other side to toast.
Serve the soup with 1 or 2 croutons on top.