Preheat the oven to 375 degrees F (190 degrees C).
Mix plums, brown sugar, confectioners' sugar, flour, cinnamon, vanilla extract, and nutmeg together in a medium bowl. Pour into the deep-dish pie crust.
Cut the regular pie crust into strips and lay over the fruit mixture. Brush milk on the crust and sprinkle sugar all over it.
Bake in the preheated oven for 40 minutes. Wrap aluminum foil over the edges of the pie to prevent burning; return to the oven and bake 7 to 10 minutes more.
