Preheat your oven to 450ºF (230ºC). Arrange the quartered tomatoes, cut side up, on a large cookie sheet or two 9x13-inch baking dishes in a single layer. Drizzle with 2 tablespoons of olive oil, then sprinkle with 1/2 teaspoon salt and a few grinds of black pepper. Roast in the oven for 30-35 minutes, until the tomatoes are soft and beginning to caramelize. Remove from the oven and let cool slightly.
While the tomatoes are roasting, heat 2 tablespoons of olive oil in a large pot over medium-high heat. Add the diced onion and sauté, stirring often, until the onion is soft and translucent, about 5-7 minutes. Then add the minced garlic and cook for another minute, stirring to avoid burning the garlic. I find that sautéing the onions until they are fully softened adds a subtle sweetness to the soup.
Add the roasted tomatoes (including the juices, peels, and seeds) from Step 1 to the pot with the sautéed onion and garlic. Pour in the broth, then add the dried basil, Italian seasoning, 1/2 teaspoon salt, and the sugar. Stir everything together well to combine all the flavors.
Using an immersion blender, blend the soup in the pot until smooth. If you don’t have an immersion blender, carefully transfer the soup in batches to a blender and blend until smooth, then return to the pot. Bring the blended soup to a gentle simmer over medium-low heat and cook for about 30 minutes, allowing the flavors to meld together. Stir occasionally to prevent sticking.
Remove the soup from the heat. Stir in the heavy cream until fully incorporated and the soup becomes silky and rich. Taste and adjust seasoning with extra salt or pepper if needed. For a creamier soup, I sometimes add an extra splash of heavy cream at the end.
Ladle the hot soup into bowls. Garnish with fresh basil leaves and a generous sprinkle of parmesan cheese. Serve immediately, ideally with grilled cheese sandwiches and a green salad for a complete meal.