Heat oven to 375°F with a rack in the lower third. Place a sheet pan on the rack to preheat.
Fit the chilled dough into a 9-inch pie plate, crimp the edge, and refrigerate while you prepare the filling.
Toss sliced apples with lemon juice, 2 tablespoons of the granulated sugar, and half the cinnamon.
Warm 1 tablespoon melted butter in a skillet over medium heat. Cook apples 4–5 minutes until slightly tender. Cool 5 minutes.
Whisk remaining granulated sugar, brown sugar, flour, remaining cinnamon, nutmeg, and salt in a bowl.
Whisk in beaten eggs, buttermilk, vanilla, and the remaining 3 tablespoons melted butter until smooth.
Spread apples in the crust. Sprinkle coarse sugar over apples, then pour the custard on top. Tap the pan to settle.
Bake on the preheated sheet pan 50–55 minutes, tenting the edge if browning quickly, until the center quivers slightly.
Cool on a rack at least 2 hours. For clean slices, chill 1 hour before serving.