Make the Brown Butter: Place the 1/2 cup of butter for the filling in a light-colored saucepan over medium heat. Melt the butter completely. Keep cooking, swirling the pan occasionally until milk solids turn golden brown and it smells nutty. Remove from heat immediately and pour into a heatproof bowl to cool.

Prepare the Pie Crust: In a large bowl, whisk together the flour and salt. Cut in the cold butter and vegetable shortening until the mixture resembles coarse crumbs. Gradually add ice water, mixing gently until the dough just comes together. Form into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Preheat Oven and Prepare Pie Pan: Preheat oven to 375°F (190°C). Lightly grease a 9-inch pie pan.

Roll Out the Pie Crust: On a lightly floured surface, roll out the chilled pie crust into a 12-inch circle. Transfer to the pie pan, trim and crimp the edges.

Make the Apple Filling: In a large bowl, combine sliced apples, cooled brown butter, granulated sugar, brown sugar, flour, cinnamon, nutmeg, and salt. Toss gently. Add lemon juice and toss again.

Assemble the Crumble Topping: In another bowl, combine flour, brown sugar, cinnamon, and salt. Cut in cold, cubed butter until crumbly. Stir in rolled oats.

Fill and Top the Pie: Pour apple filling into the pie crust. Sprinkle crumble topping evenly over the apples.

Bake the Pie: Bake for 45-55 minutes, or until the crust is golden brown and filling is bubbly. Tent with foil if topping browns too quickly. Let cool on a wire rack for at least 20 minutes before serving.