Peel, core, and thinly slice the apples. Place them in a large mixing bowl and add lemon juice, brown sugar, granulated sugar, cinnamon, nutmeg, vanilla extract, and flour. Toss everything until the apple slices are evenly coated. Set aside for approximately 15 minutes to allow the flavors to combine.
In a large mixing bowl, beat the softened butter and granulated sugar until light and creamy. Add the egg and vanilla extract, mixing until combined.In another bowl, whisk together the flour, cream of tartar, baking soda, salt, and cinnamon. Gradually add the dry ingredients to the butter mixture. Mix until a soft dough forms. Divide the dough into two portions, with one slightly larger than the other. Wrap each portion and refrigerate for approximately 30 minutes. Chilling helps the dough become easier to handle and prevents excessive spreading during baking.
Preheat the oven to 350°F (175°C). Lightly grease a 9-inch pie dish.Roll out the larger portion of chilled dough between two sheets of parchment paper. Carefully transfer it into the pie dish, pressing gently along the bottom and sides. Trim any excess dough. Pour the prepared apple mixture into the crust, arranging the slices evenly. Distribute the small pieces of butter over the filling. If the apples have released excess liquid, avoid pouring all of it into the pie.
Roll out the remaining dough and place it over the apple filling. You can cover the pie completely, create a lattice pattern, or use decorative cookie-dough shapes.Seal the edges by gently pressing the top and bottom crusts together. Cut a few small vents into the top crust to allow steam to escape. Combine granulated sugar and cinnamon in a small bowl. Brush the top crust with melted butter and sprinkle the cinnamon-sugar mixture evenly over it. Add coarse sugar if desired for extra crunch.
Place the pie on a baking sheet and bake for 50–60 minutes, or until the crust is golden brown and the apple filling is bubbling through the vents. If the edges brown too quickly, cover them loosely with aluminum foil.
Remove the pie from the oven and allow it to cool for at least 2–3 hours. This helps the filling set properly before slicing. Serve at room temperature or gently warmed with vanilla ice cream, whipped cream, or caramel sauce.