**Preheat Oven:** Preheat your oven to 375°F (190°C).
**Prepare the Filling:** In a large mixing bowl, combine the apples, strawberries, granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, salt, lemon juice, and vanilla extract. Toss until evenly coated.
**Assemble the Pie:** Roll out one pie crust and place it into the pie dish, allowing the edges to hang slightly over the sides. Pour the fruit filling into the crust, spreading it evenly.
**Top the Pie:** Roll out the second pie crust and place it over the filling. Trim and crimp the edges to seal. Use a fork or your fingers to press them together. Cut slits in the top crust to allow steam to escape.
**Egg Wash and Sugar:** Brush the top crust with the beaten egg and sprinkle with granulated sugar for a golden, crispy finish.
**Bake:** Place the pie on a baking sheet to catch any drips. Bake in the preheated oven for 45-50 minutes, or until the crust is golden brown and the filling is bubbling.
**Cool:** Allow the pie to cool for at least 2 hours before slicing to let the filling set.