Melt butter in a large saucepan over medium heat. Cook and stir onion and mushroom in hot butter until onion is tender, about 3 minutes. Stir flour, pepper, and salt into the onion mixture. Gradually stream milk and chicken broth into mixture while stirring to incorporate.
Bring the liquid to a boil; cook at a boil until thickened, about 2 minutes. Remove saucepan from heat. Stir parsley and nutmeg into the soup. Ladle soup into bowls and top each with a dollop of sour cream.