In a food processor purée the tomatoes and force the purée through a fine sieve into a saucepan, pressing hard on the solids.
In a small bowl stir together the cornstarch and the broth and stir the mixture into the tomato purée.
Bring the mixture to a boil, stirring, remove the pan from the heat, and stir in the lemon juice, the sugar, the whole basil leaves, and salt and pepper to taste.
Let the soup cool and chill it, covered, for at least 8 hours.
The soup may be made 2 days in advance and kept covered and chilled.
Discard the whole basil leaves, ladle the soup into 6 bowls, and garnish each serving with a dollop of the sour cream and some of the chopped basil.
Drizzle the soup with the oil and serve it with the toasts.
