Peel, core and slice apples. Combine with lemon juice and 1/2 cup sugar. Chill. Drain apple mixture and return 1/4 cup liquid to mixture.
Stir in flour, cinnamon, nutmeg, brown sugar, and rum. Chill.
Prick bottom and sides of pie shell with a fork. Spoon apple mixture into prepared pie crust, forming a dome.
Cover with second pie crust. Make slits in pie crust. Brush with milk and sprinkle with sugar. Cover edges with aluminum foil. Bake on a baking sheet at 375 degrees for 25 minutes. Remove foil and bake for 20 - 35 minutes.