Combine milk, sugar, and butter together in a 2-quart saucepan over medium-high heat; cook and stir until butter melts, 3 to 5 minutes. Pour about 1 cup milk mixture into a bowl and whisk cornstarch into mixture until smooth. Pour milk-cornstarch mixture back into the saucepan. Cook, stirring constantly, over medium heat until thickened, 5 to 10 minutes.
Pour milk mixture into the pie crust and sprinkle with nutmeg. Refrigerate pie until firm, about 3 hours.