First, to get the skins off, just dunk them in boiling water for about a minute. See how the skins just slide off. I thought skinning all those little guys was goint to be a daunting task, but I was done in about ten minutes.
Then, in a large stock pot, heat 1 T olive oil over medium-high heat and add 1 medium chopped onion. Cook and stir occasionally, until softened.
Tie 6 sprigs of basil together with kitchen twine. Put peeled tomatoes in the pot with the onion, 4 cups chicken stock, 1 1/2 t salt, 1/2 t pepper and the basil. Bring to a boil over high heat, then reduce to medium low and let simmer about 20 minutes. Stir occasionally to break up tomatoes. Let cool. Remove basil sprigs.
In a blender, puree the soup in batches until very smooth. (Now, if you're inpatient and don't let it cool completely, you've got to do it in very small batches - like 1 cup, and hold the lid down very tightly. I did this with carrot soup once; it blew the top of the blender, and the kitchen and I were covered in carrot soup. It's a mistake you'll only make once, but once is one time too many.)
Strain the soup into a clean pot, add about 2 oz. of sharp chedder cheese, shredded, and bring to a simmer over medium heat. Remove from heat and stir in 1 cup heavy cream and 2 t balsamic vinegar. Add salt and pepper to taste.
Ladle the hot soup into bowls and garnish with slice basil. Makes about 8 cups.