Combine 3 cups of water and the white rice in a pot.

Simmer, covered, until the rice is very soft, about 20 minutes.

Heat the vegetable oil in a saucepan over low heat.

Add the chopped onion and cook until slightly soft.

Add the fresh peas and toss for 1 minute.

Add the cooked rice with its cooking liquid, sugar, and salt.

Simmer, covered, until the peas are tender, about 8 to 10 minutes.

Pass the soup through the medium disk of a food mill.

Slice the fresh sorrel into slivers.

Reserve a handful of sliced sorrel for garnish.

Place the rest of the sorrel in a nonaluminum pot with the remaining 1/2 cup water.

Stir over moderate heat until the sorrel is melted.

Add the melted sorrel to the pureed soup.

Press the mixture through a fine sieve.

Add 1/2 cup buttermilk and 1/4 cup heavy cream.

Adjust seasoning with salt and white pepper to taste.

Refrigerate the soup for at least 4 hours to chill thoroughly.

Before serving, add more buttermilk and cream to taste and check seasoning.

Ladle the chilled soup into bowls.

Garnish with the reserved sorrel slivers and serve immediately.