Preheat your grill to 400 degrees. 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.
Heat the remaining olive oil over medium heat in a large, heavy bottomed pot. Stir in the carrots and onions and saute 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.
Slowly transfer the roasted tomatoes from the baking sheet into the soup pot, cover with the lid and simmer for 20 minutes.
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.
Pour the remaining 1/4 cup of heavy cream into a bowl and beat until whipped and soft peaks form. Mince the remaining 1/4 cup of basil and fold into the whipped cream.
Serve the warm soup with a spoonful of the basil cream floating on top and toasted bread or grilled cheese sandwiches.