Wash, peel, core, and thinly slice the apples. Place them in a large mixing bowl and add the lemon juice. Toss gently to coat the apple slices and prevent browning.
Add brown sugar, granulated sugar, flour, cinnamon, cardamom, ginger, cloves, nutmeg, and black pepper to a small bowl. Mix until the spices are evenly combined.
Sprinkle the spice mixture over the sliced apples. Add vanilla extract and gently toss until every apple slice is coated. Allow the mixture to rest for approximately 10 minutes so the flavors can blend.
Preheat your oven to 400°F (200°C). Place one pie crust into a 9-inch pie dish. Gently press it against the bottom and sides of the dish, leaving a little excess pastry around the edges.
Transfer the prepared apple mixture into the pie crust. Arrange the slices evenly and pile them slightly higher in the center. Cut the butter into small pieces and distribute them over the filling.
Whisk the egg and milk together in a small bowl. Brush the mixture gently over the top crust. Sprinkle coarse sugar over the surface if desired for a beautiful golden finish.
Place the pie on a baking sheet to catch any bubbling filling. Bake at 400°F (200°C) for 20 minutes. Reduce the oven temperature to 350°F (175°C) and continue baking for another 35–45 minutes, or until the crust is golden brown and the apple filling is bubbling.
Remove the pie from the oven and place it on a cooling rack. Allow it to cool for at least 2 hours before slicing. This helps the filling thicken and makes the slices easier to serve.