In a large pot or Dutch oven, combine the shelled peas, water, chopped onion, parsley sprigs, 1 teaspoon (5 ml / 0.2 fl oz) of salt, and sugar.
Bring the mixture to a boil over high heat, then reduce the heat to low, cover, and let it simmer gently for 40 minutes.
While the soup simmers, wash and coarsely chop the lettuce.
In a separate skillet, melt 3 tablespoons (45 ml / 1.5 fl oz) of the butter over medium heat.
Add the chopped lettuce and sauté, stirring occasionally, until it is completely wilted and soft, and most of its liquid has evaporated.
Carefully transfer the cooked lettuce and any butter from the skillet into the simmering soup pot.
Continue to simmer for an additional 10 minutes.
Carefully transfer the soup to a blender in batches, ensuring not to overfill and venting the lid to allow steam to escape.
Blend until completely smooth and velvety. Alternatively, use an immersion blender directly in the pot.
Return the puréed soup to the pot over low heat.
Stir in the dry white wine, heavy cream, and the remaining 1 tablespoon (15 ml / 0.5 fl oz) of butter.
Heat gently until the soup is warmed through, but do not let it boil.
Taste and adjust the seasoning with additional salt and a dash of white pepper as needed.
Serve hot, garnished with small soup dumplings if desired.
