Preheat oven to 300°F.
Place tomatoes, quartered onion, carrots, and garlic in a Dutch oven. Drizzle with olive oil;
sprinkle with 2 teaspoons salt and 1 teaspoon pepper. Toss to coat. Add 1 cup water to pan.
Bake, uncovered, 1 hour. Cover, and bake until vegetables are tender, about 1 hour more.
Remove pan from oven. Using a potato masher, crush vegetables until mixture reaches a chunky texture.
Stir in crushed tomatoes, tomato paste, and stock, if desired. Bring to a simmer over medium heat. Cook, stirring occasionally, to let flavors meld, about 15 minutes.
If desired, use an immersion blender to puree soup to desired texture. Stir in half and half; season with remaining 1/2 teaspoon salt and 1/2 teaspoon pepper to taste.
Taste soup; stir in desired amount of brown sugar and balsamic vinegar.
Garnish servings with basil leaves and grated Parmesan. Serve with garlic bread or your favorite grilled cheese sandwich.