How to Make It
In a large pot, heat the olive oil. Add the parsnip, leeks, celery, onion, garlic, tarragon and bay leaf and cook over moderate heat until the vegetables have softened, about 5 minutes. Add the chicken stock, potatoes and 1/2 teaspoon each of salt and white pepper and bring to a boil. Cover and cook over moderately low heat for 45 minutes. Discard the bay leaf.
Add the peas to the pot and simmer until tender, about 2 minutes. Remove from the heat and stir in the heavy cream.
Working in batches, puree the soup in a blender until very smooth. Strain the soup through a fine sieve set over a large bowl, pressing on the solids. Press a piece of plastic wrap directly on the surface of the soup and refrigerate for at least 8 hours or overnight.
Season the chilled soup with lime juice, salt and white pepper. Garnish with the chives and mint and serve.