For the soup: Heat the butter and olive oil in a braiser or shallow Dutch oven over medium heat. Add the garlic, onion, thyme and oregano. Season with a few pinches of salt and pepper. Cook until the onions are translucent, about 5 minutes.
Increase the heat to medium high and add the tomato paste. Stir and cook for 30 seconds. Add the wine, scraping the bottom of the pot to deglaze. Pour in the tomato juice, stewed tomatoes and sugar. Bring it to a boil and reduce to a simmer. Simmer gently for 10 minutes.
For the croutons: Meanwhile, drizzle both sides of each baguette slice with 1/2 tablespoon of the olive oil and arrange them on a baking sheet. Bake until slightly toasted, 6 to 8 minutes. Set aside.
Add the heavy cream, Parmesan, pecorino and basil to the soup. Taste and adjust the seasoning if necessary.
To serve, preheat the broiler. Ladle the soup into 6 ovenproof bowls and put on a baking sheet. Place 2 croutons onto each bowl and top with a piece of the Cheddar. Broil the soups just long enough for the cheese to get bubbly and slightly toasted. Serve each bowl with a sprinkle of parsley.