Heat oil or butter in a large pot or Dutch oven. Sauté onion on low heat until softened. Add peas, mint, parsley, and broth, stirring well. Bring to a boil, then reduce heat and simmer, partially covered, for 20 minutes.

Working in batches, purée in a blender until smooth. Again working in batches, press soup through a fine sieve, discarding pea fibers. Add buttermilk to strained soup, plus salt and pepper to taste. Chill. Serve with a sprig of mint.