Prepare the pastry or set out two refrigerated 9-inch pie crusts. If making pastry from scratch, chill the dough for at least 30 minutes before rolling.

Peel, core, and slice six to eight Granny Smith apples into pieces about 1/4 inch thick. Toss the slices with lemon juice.

Combine the granulated sugar, brown sugar, flour, cinnamon, nutmeg, and salt. Toss the mixture with the apples until every slice is coated.

Fit one crust into a 9-inch pie plate and add the apples, mounding them slightly in the center. Dot the filling with the two tablespoons of butter.

Cover the apples with the second crust, trim and crimp the edges, and cut several vents in the top. Brush with the egg wash if desired.

For the optional caramel-style sauce, melt the butter, whisk in the flour, then add the water and both sugars. Simmer briefly until smooth, then pour some through the vents before sealing or baking, depending on your preferred recipe version.

Bake at 425°F for 15 minutes, then reduce the oven to 350°F and bake for another 35 to 45 minutes. The pie is ready when the crust is golden, juices bubble through the vents, and the apples can be pierced easily with a knife.

Cool the pie for at least two hours before slicing. This resting time allows the fruit juices to thicken so the filling stays in the slice instead of running across the plate.