Place a pizza stone on the lower-middle oven rack with foil on the rack below to catch drips. Preheat oven to 425°F for at least 30 minutes.
Place sliced apples in a large bowl. In a small bowl, whisk granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, and salt. Toss with the apples along with vanilla, almond extract, and water until evenly coated. Let rest 20 minutes to draw out excess moisture.
Fit one pie crust into the pie plate. Using a slotted spoon, transfer apples into the crust, leaving liquid behind in the bowl. Mound apples slightly higher in the center and dot with butter.
Top with the second crust, trim excess, fold edges under, and crimp. Cut 4-5 slits in the top to vent steam.
Whisk egg with water and brush over the top crust and edges. Sprinkle with coarse sugar, if using.
Place pie directly on the preheated stone. Bake at 425°F for 20 minutes, then reduce to 375°F (without opening the oven) and bake 40-50 minutes more, until deep golden brown and filling bubbles through the vents. Tent edges with foil if browning too quickly.
Cool on a wire rack at least 3-4 hours before slicing to allow the filling to set.
