Drain the split peas, reserving the cooking liquid.
Blend the split peas until you have a smooth, velvety soup, adding as much of the cooking liquid as needed.
Personally, I ended up using all of the cooking liquid. I recommend not throwing away the unused water. First, because it’s very tasty and nutritious (you can cook it with broth and vermicelli, for example). But also, split peas tend to thicken once they sit, and a soup that’s thin at first can become pasty an hour later!