Preheat the oven to 400℉. Drizzle half of an onion and garlic cloves in olive oil, balsamic vinegar, salt, and black pepper. Wrap loosely in foil, place in a roasting pan or cast iron skillet, and roast for 30 minutes. Once roasted, remove the garlic cloves from the garlic skin and pour the roasted garlic, onion, oil, and vinegar into the roasting pan.
Remove the crust from the slice of sourdough and break the bread into crumbs. Add the tomatoes, Italian seasoning, paprika, red pepper flakes, parsley, basil, Italian seasoning, chicken broth, and sourdough crumbs to the roasted garlic/ onion mixture. Roast for another 45 minutes until the soup is bubbly, and slightly reduced.
Using a blender or immersion blender, blitz the soup until it is completely smooth and creamy. Return to the pan. Add the cheese and stir until melted and fully combined. If you want to make sure the cheese is fully combined, blend it one more time. Add 2 tablespoons of butter and still til melted. Add salt and pepper to taste. Enjoy with some grilled cheese sandwiches or cheesy toast.