Preheat the oven to 425°F and set aside a 9-inch pie plate.
Melt the butter in a large pot over medium-low heat.
Add apples, brown sugar, flour, cinnamon, lemon juice, and salt. Stir and cook for 8–10 minutes until the apples begin to soften. Remove from heat and let cool.
Divide the pie dough into two discs if needed. On a lightly floured surface, roll one disc to about 1/8-inch thick and slightly larger than the pie plate.
Transfer the dough to the pie plate and gently press into the bottom and sides.
In a small bowl, whisk together the egg and water to make the egg wash.
Spoon the cooled apple filling into the prepared crust.
Roll out the second dough disc and place over the pie, or cut into strips for a lattice or braided top.
Trim excess dough and crimp the edges to seal. Cut vents in the top crust if not using a lattice.
Brush the top with egg wash and sprinkle with coarse sugar if desired.
Bake for 15 minutes at 425°F. Reduce oven temperature to 350°F and continue baking for about 40 minutes, until the crust is golden brown and filling bubbles through the vents.
Cover edges loosely with foil if the crust browns too quickly. Cool before serving.