Step 1: Preheat oven to 375°F (190°C). In a large bowl, whisk together the flour, granulated sugar, and salt for the crust.
Step 2: Add the cold, cubed butter to the flour mixture. Using a pastry cutter or your fingertips, work the butter into the flour until it looks like coarse crumbs with pea-sized pieces of butter still visible.
Step 3: In a separate small bowl, beat the eggs, then add the cold water and mix well.
Step 4: Gradually pour the egg mixture into your flour and butter mixture, stirring gently until the dough just starts to come together. If it feels too dry, add more cold water, one tablespoon at a time, until you can form it into a ball. Do not overmix.
Step 5: Divide the dough into two equal parts, wrap each in plastic wrap, and chill in the fridge for at least 30 minutes.
Step 6: While the dough is chilling, prepare the apple filling. In a large bowl, combine the thinly sliced apples, granulated sugar, light brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss to ensure every apple slice is coated.
Step 7: Once the dough is chilled, grab one portion and roll it out on a lightly floured surface into a rectangle, aiming for about 1/4 inch thick. Carefully transfer this dough to a greased or parchment-lined baking sheet.
Step 8: Spread the apple filling evenly over the bottom crust. Drizzle the melted butter on top.
Step 9: Roll out the second portion of dough to the same size as your first crust and gently lay it over the apples. Pinch the edges of both crusts together to seal. Use a sharp knife to cut a few small slits on top to let steam escape.
Step 10: Bake in the preheated oven for 40-45 minutes, or until the crust is beautifully golden brown and the apple filling is bubbling around the edges.
Step 11: While your apple slices are cooling a bit, prepare the glaze. In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until you have a smooth, pourable glaze. Adjust consistency with more milk or powdered sugar if needed.
Step 12: Once the apple slices are still warm (but not piping hot!), drizzle the sweet glaze generously over the top. Then, slice and serve immediately.