Preheat. Preheat your smoker to 400 degrees. I used my pellet smoker for this recipe, but you can use your favorite smoker or any grill with an indirect heat source.
Smoke the tomatoes. Place the quartered tomatoes on a large baking sheet and drizzle with 1 Tablespoon of the olive oil. Set the baking sheet on the grill grates, close the lid, and roast for 30 minutes.
Make the soup base. Heat the remaining olive oil over medium heat in a large, heavy bottomed pot. Stir in the carrots and onions and sauté for about 10 minutes, until the vegetables are tender. Stir in the garlic and cook for an additional 2 minutes. Pour in the chicken stock, tomato paste, sugar, salt, pepper, and 1/8 cup of the fresh basil. Bring up to a simmer.
Add the tomatoes. Slowly transfer the roasted tomatoes from the baking sheet into the soup pot, cover with the lid and simmer for 20 minutes.
Puree and finish cooking. Carefully ladle the soup mixture into a blender and puree in batches until all of the soup is pureed. Pour the blended soup through a strainer to remove any leftover chunks or pulp. Return the soup to the pot and stir in 1/2 cup of the heavy cream. Heat to desired temperature for serving.
Make the basil whipped cream. Pour 1/4 cup of heavy cream into a bowl and beat until whipped and soft peaks form. Mince the 1/4 cup of basil and fold into the whipped cream.
Serve. Serve the warm soup with a spoonful of the basil cream floating on top and toasted bread or grilled cheese sandwiches.