Preheat the oven to 375°F (190°C).
In a mixing bowl, whisk together the flour, sugar, and salt for the pie crust. Add the cold, diced butter and mix until crumbly.
Form the dough into a ball, then flatten it into a disc. Wrap in plastic wrap and refrigerate for 15 minutes.
Roll out the chilled dough on a floured surface to fit the pie dish. Place the rolled dough in the dish, trim the edges, and poke holes in the bottom with a fork.
Brush the crust with a beaten egg for a golden finish.
In a large bowl, combine the sliced apples, sugar, cinnamon, nutmeg, lemon juice, and flour. Toss until the apples are evenly coated.
Pour the apple mixture into the prepared crust, spreading it evenly.
In another bowl, combine the oats, brown sugar, flour, cinnamon, and salt for the crumble topping. Add the cold, diced butter and mix until crumbly.
Sprinkle the crumble topping evenly over the apple filling.
Place the pie on a baking sheet and bake in the preheated oven for about 45 minutes, or until the apples are tender and the topping is golden brown.
Allow the pie to cool for at least 15 minutes before slicing and serving.