Preheat the oven to 400 degrees. Coat a baking sheet with olive oil cooking spray.
Roast the tomatoes by straining them of juice, reserving the juice for later. Add tomatoes to the baking sheet, and drizzle 2 tablespoons of olive oil over the top along with dried basil, sea salt, and pepper to taste.
Slide the garlic slivers into the tomatoes; they will roast and soften inside & infuse the tomatoes with their flavor.
Place the baking sheet in the oven and roast for approximately 20-25 minutes. Side Note: Don't over-roast or they will be dry.
Prepare the soup by heating 1 tbsp of olive oil in a Dutch oven over medium heat. Add onion, carrot, and celery, and cook until tender, about 5 minutes.
Add garlic and stir for 30-45 seconds.
Stir in the broth, reserved tomato juice (from canned tomatoes), bay leaf, sugar, balsamic vinegar, and the roasted tomatoes. Cover with a lid and cook over low heat for 2-3 hours, stirring occasionally.
Temper the heavy cream with a bit of the soup and mix well. Add the tempered heavy cream and the fresh basil to the soup.
Remove the bay leaf and use an immersion blender to blend until smooth. Side Note: If you want a thin soup, pour the mixture through a strainer. If you want a thicker soup, serve it as is.