Bring water to a boil in a large pot. Add peas, return to a boil and then reduce to a simmer. Cook, stirring occasionally, for 45 minutes.

Using a slotted spoon, transfer one-third of the pea pods to a food processor. Add 1/2 cup cooking liquid and process until smooth. (Use caution when pureeing hot liquids.) Pour into a large bowl. Repeat with the remaining pea pods in 2 batches, with 1/2 cup cooking liquid each time. Pour the pureed peas plus the remaining cooking liquid through a fine-meshed sieve, pressing on the solids to extract as much liquid as possible. (Alternatively, put through a food mill fitted with a fine disc.)

Return the soup to the pot, bring to a boil and then simmer until reduced by about a third (to about 6 cups), 30 to 35 minutes. Stir in chopped dill, salt and pepper. Ladle into bowls and top each serving with a swirl or dollop of yogurt and a sprig of dill, if desired.