Unroll 1 pie crust on a work surface; brush with water. Place second pie crust atop the first pie crust; press gently together. Place the double crust in a 9-inch pie pan.
Beat cream cheese, 1/2 cup sugar, 1 egg, vanilla, and salt together in a bowl until smooth and creamy. Pour into pie crust; sprinkle cheesecake filling with pecans.
Mix corn syrup, butter, 1/4 cup sugar, 3 eggs, and rum extract together in a bowl until smooth; pour over pecan layer.
Bake in the preheated oven until a knife inserted in the center of the pie comes out clean, 45 to 50 minutes.
