Prepare your pie crust dough and refrigerate for at least 1 hour before assembling. Preheat oven to 425°F with a rack in the center.
Melt butter in a medium saucepan over medium heat. Add flour and whisk constantly for 1 minute. Whisk in water and sugar, simmer for 3 minutes, then remove from heat immediately.
Peel, core, and slice apples into ¼-inch slices. Sprinkle with cinnamon, toss to coat, then stir in the warm filling sauce until all slices are well coated.
Roll the first dough disk into a 12-inch circle and transfer to a 9-inch deep-dish pie pan. Add the apple filling, mounding slightly in the center, keeping filling away from the edges.
Roll the second dough disk into an 11-inch circle and cut into 10 strips with a pizza cutter. Weave strips into a lattice pattern over the filling. Crimp edges firmly to seal, then brush the entire top with egg wash.
Bake at 425°F for 15 minutes, then reduce to 350°F and bake an additional 45 minutes, until thick juices bubble through the lattice for at least 5 minutes. Tent edges with foil if browning too fast.
Remove from oven and cool on a wire rack for at least 1 hour before slicing.