Combine milk, breadcrumbs, and egg in a medium saucepan; stir until well blended. Cook over low heat, stirring frequently, until mixture thickens. Remove from heat, and stir in salt, nutmeg, and pepper. Cool slightly.
Combine cooled sauce and chicken in container of an electric blender; process until smooth.
Spoon mixture into a pastry bag fitted with a 1/4-inch plain tip. Pipe about 1 tablespoon chicken mixture into a small "S" shape in a large buttered skillet. Repeat procedure with remaining mixture, spacing quenelles 1 inch apart, until surface of skillet is covered.
Pour hot chicken broth very slowly down the sides of skillet so as not to disturb quenelles, adding enough broth to cover quenelles. Cover and simmer 4 minutes or until quenelles float to surface of broth. Remove cooked quenelles with a slotted spoon; drain on paper towels. Pour off broth, and set aside. Repeat procedure with remaining chicken mixture and reserved broth.
Add quenelles to remaining hot chicken broth or other broths or consommés before serving.