In a large bowl, mix flour, sugar, and salt. Cut in cold butter with a pastry cutter until crumbly. Add ice water gradually until the dough comes together. Divide in half, shape into discs, wrap, and chill for at least 1 hour.

Toss sliced apples with sugars, flour, cinnamon, nutmeg, salt, lemon juice, and vanilla. Let sit while preparing the crust.

Roll out one crust and place into a deep-dish pie plate. Fill with apple mixture, mounding high in the center.

Roll out the second crust and place over the apples. Trim, crimp edges, and cut vents or create a lattice top.

Whisk egg and milk, then brush over crust. Chill pie in freezer for 20 minutes before baking.

Bake at 400°F (200°C) for 20 minutes. Lower heat to 375°F (190°C) and continue baking for 45–55 minutes until golden brown and bubbling.

Optional: While pie bakes, make caramel by heating sugar until melted and amber. Stir in butter, then cream and salt. Drizzle over warm pie before serving.