Preheat your oven to 425°F (220°C). Arrange halved Roma tomatoes, cherry tomatoes, and quartered onion on a large rimmed baking sheet. Drizzle with 3 tablespoons of olive oil, balsamic vinegar, dried oregano, kosher salt, and black pepper. Place the head of garlic cut-side up on a square of foil, drizzle with oil, wrap tightly, and nestle onto the sheet. Roast for 25 minutes until tomatoes are blistered, tender, and slightly charred at the edges.

Transfer all the roasted tomatoes, onions, and pan juices directly into a large soup pot or Dutch oven. Squeeze the soft roasted garlic cloves out of their skins into the pot (discard skins).

Pour in chicken bone broth and add the packed fresh basil leaves. Using an immersion blender (or transferring in batches to a high-speed blender), purée the soup until completely smooth, silky, and velvety.

Place the pot over medium-low heat. Stir in heavy whipping cream and freshly grated Parmigiano-Reggiano. Simmer gently for 5 minutes to marry all flavors. Taste and adjust seasoning with additional salt or cracked pepper if needed.

While soup simmers, butter sourdough slices on one side and sandwich with shredded cheddar and Swiss. Grill in a skillet over medium heat until golden-crusted and cheese is gooey. Cut into triangles. Ladle hot soup into bowls, swirl with cream, garnish with fresh basil, and serve with warm grilled cheese dippers.