In a large heavy stockpot or casserole, melt the butter until it foams. When the foam subsides, add the onions and cook over moderate heat until softened but not browned, about 5 minutes. Add the turnips and potatoes and stir to coat with the butter. Add 2 teaspoons of salt, cover and cook over low heat, stirring occasionally, until tender, about 20 minutes.
Stir in the chicken stock and bring to a simmer. Cover partially and cook over moderate heat until the vegetables are very tender, about 10 minutes.
Working in batches, puree the soup in a blender until perfectly smooth. Return the soup to the pot and season with salt and the nutmeg. Ladle the soup into shallow bowls and garnish with the basil before serving.