Preheat oven to 425 degrees F (220 degrees C). Line the bottom of a deep-dish pie plate with 1 pie crust.
Combine apples, brown sugar, molasses, corn syrup, cornstarch, rum extract, vanilla extract, cinnamon, and nutmeg in a bowl. Mix well and pour over the crust. Cover with top crust. Cut slits into the dough and crimp the edges. Wrap bottom and sides of the pie plate in aluminum foil.
Bake in the preheated oven until crust is golden brown, 50 minutes to 1 hour. Cool for 15 to 20 minutes and serve with caramel syrup drizzled on top.