Prepare pie crust and chill until needed.  

Cut a small slice off of the tops of six medium apples and remove the stem area. Using a melon baller, hollow out the apple removing the core and some of the apple; leaving approximately a 3/4-inch wall.  

Peel and core 4 to 5 large apples, dice in medium to small cubes, set aside.  

In a large skillet, melt butter on medium heat. Add sugars, lemon juice and cinnamon. Stir in cubed apples. Stir frequently and cook 5 to 6 minutes. Add in cornstarch/water mixture; stir well.  

Turn off heat and stir in walnuts, raisins and rum. Fill hollowed out apples with mixture. Set aside to cool.  

Preheat oven to 400°F.  

Roll out the pie crust to approximately 1/4-inch thickness.  

For lattice topping: Cut dough into 1/2 x 5-inch strips. Form lattice top; lift and place over the filled apple. Press down on sides of apple, trim carefully with dull knife. Brush on egg wash and sprinkle with sugar.  

For solid topping: Cut dough into a circle and drape over the top of the filled apple. Press dough on sides of apple, cut vent hole and brush on egg wash.  

Place apples in a 13 x 9-inch pan. Bake 10 to 14 minutes until the dough is lightly golden brown. Remove from oven and cool slightly. Serve warm.