Set a rack in the lower third of the oven and heat to 425°F (220°C). Place a rimmed baking sheet on the rack to catch bubbling juices.
Toss the apple slices with both sugars, flour, cinnamon, nutmeg, salt, and lemon juice. Let stand for 10 minutes while you fit the bottom crust into a 9-inch pie plate.
Spoon the filling into the crust, mounding it slightly in the center. Dot with butter. Roll the second crust and cut it into strips; weave the strips over the apples and crimp the edges.
Brush the pastry with egg wash and sprinkle with coarse sugar if using. Bake on the hot sheet for 20 minutes. Reduce the oven to 375°F (190°C) and bake 35 to 45 minutes more.
If the edge browns too quickly, shield it with foil. The pie is ready when the crust is deep golden and juices bubble through the lattice. Cool on a rack for at least 2 hours before slicing.