In a large bowl, whisk together flour, sugar, and salt. Cut in the cold butter using a pastry blender until mixture resembles coarse crumbs with pea-sized pieces of butter remaining.

Gradually sprinkle in the ice water, mixing gently with a fork until the dough just begins to come together. Divide the dough into two discs (one slightly larger), wrap in plastic, and chill for at least 1 hour.

In a large bowl, toss the peeled and sliced apples with lemon juice. In a separate small bowl, whisk together the brown sugar, granulated sugar, flour, and cinnamon. Sprinkle the sugar mixture over the apples and toss to coat evenly.

Preheat oven to 400°F (200°C). Roll out the larger dough disc and fit it into a 9-inch pie dish. Spoon in the apple filling and dot with butter pieces.

Roll out the second dough disc and cut into ten 1-inch strips. Weave the strips into a lattice pattern over the filling. Trim and crimp the edges.

Brush the lattice with an egg wash (1 egg whisked with 1 tbsp milk). Sprinkle with a mixture of coarse sugar and cinnamon.

Bake on a baking sheet at 400°F (200°C) for 20 minutes. Reduce heat to 375°F (190°C) and bake for another 35-45 minutes, until the crust is deep golden-brown and the filling is bubbly. Let cool completely (at least 4 hours) before slicing.