In a Dutch oven or soup pot, melt the butter. Add the onion, season with salt and pepper, and cook over medium heat, stirring frequently until the onion is translucent, about 5 to 6 minutes. Add the water and diced potato and cook, partially covered, until the potato cubes are tender, about 10 minutes. When poked with a paring knife, the knife should meet no resistance.

Turn off the heat and add the watercress and peas, stirring a few times to wilt the cress.

Puree the soup using an immersion blender, or wait until the soup is lukewarm and puree until smooth in a blender. (Pureeing hot soup in a traditional blender will cause it to overflow, so wait until the soup is tepid.)

Once the soup is pureed, rewarm the soup in the pot, season with salt and pepper, and serve.