Prepare Caramel Sauce: In a saucepan, melt butter, brown sugar, and corn syrup over medium heat. Stir until well combined, then gradually add sweetened condensed milk. Cook until bubbling and slightly thickened, about 4-5 minutes, before stirring in vanilla. Let it cool.

Make Apple Filling: Peel, core, and slice the apples into thin wedges. In a large skillet, melt butter over medium heat and add the apple slices. Cook for 15-20 minutes until they soften and begin to caramelize, adding sugars and lemon juice while cooking.

Thicken Filling: In a small bowl, dissolve cornstarch in cold water. Slowly mix this into the apple filling until it thickens slightly. Remove from heat and stir in ground cinnamon and nutmeg. Allow to cool before assembling.

Assemble Pie: Roll out the pie crust and fit it into a pie dish. Pour the cooled apple mixture into the crust, drizzle half of the prepared caramel sauce over the apples, and cover with a top layer or lattice design.

Bake: Cut vent holes in the top crust and brush with egg wash. Sprinkle sugar on top and bake in a preheated 400°F oven for about 60 minutes until the crust is golden brown and filling is bubbling.

Cool: Allow the pie to cool for at least 1 hour before slicing to let the filling set.