Preheat the oven to 400?F. Line a baking sheet with foil. Add the tomatoes, 1/4 cup of the olive oil, garlic cloves, red pepper flakes, oregano, thyme, basil, kosher salt, and ground black pepper to the pan. Toss until the ingredients are evenly coated, then position the tomatoes cut side up. Roast in the oven for 40-45 minutes until caramelized and lightly browned. Set aside.

Heat the remaining 1 1/2 tablespoons of olive oil in a pot over medium high heat. Add in the onion and carrot, and saut? for about 5 minutes. Mix in the tomato paste and cook for 1 minute. Add in the roasted tomatoes and garlic. Mix to combine, then pour in the vegetable stock.

Working in batches, transfer the mixture to a blender along with the fresh basil leaves and blend until smooth. Alternatively, use an immersion blender and blend until smooth. Pour the soup back into the pot and heat over medium heat until it comes to a simmer. Mix in the parmesan cheese and heavy cream.

Spread about 1/2 a tablespoon of the unsalted butter onto one side of each slice of sourdough. Flip the slices over and spread about 1 tablespoon of the boursin cheese onto each slice. Add about 1/4 cup of colby jack cheese and 1/4 cup of sharp American cheddar cheese to two of the slices. Top with the remaining slices, making sure the butter-coated side faces up.

Heat a pan over medium low to medium heat. Add the sandwiches to the pan and cook for about 3 minutes until the slice on the bottom is golden brown. Flip the grilled cheese over and cover the pan with a lid. Cook for another 3-4 minutes until the bottom slice is golden brown and the cheeses have melted.