1. To make crust: Combine first 5 ingredients (through oil) in a food processor and pulse until mixture resembles coarse meal. Combine the cider and water in a small bowl and drizzle on the dough, while processing, until dough is moist and begins to stick together. Cover in plastic wrap; chill 30 minutes.
2. Preheat oven to 400°.
3. To make filling: Toss apples with lemon juice, brown sugar, 3 tablespoons maple syrup, and flour.
4. Unwrap dough and place on 16-inch square of parchment paper. Roll into a 15-inch circle. Place dough and parchment on rimmed baking sheet; arrange apples in center, leaving a 2-inch border. Fold edges in; press gently to seal. Dot apples with butter.
5. Whisk egg white and water together in a small bowl. Brush edge of dough with egg wash and sprinkle with coarse sugar. Loosely cover with foil; bake additional 25 minutes. Uncover; bake 15-20 minutes or until tender and golden.
6. Let stand 20 minutes. Brush apples with remaining 1 tablespoon maple syrup. Cut into 8 wedges; serve.