Preheat oven to 350°F.
For the filling, peel and core apples, giving remaining flesh a thin slice about ⅛ of an inch thick. Place prepared apples in a large bowl and toss with lemon juice. Add sugars and toss to combine; then add the ⅓ cup flour, teaspoon salt, and cinnamon and gently toss apples to coat. Set aside.
For the pie dough, in a separate bowl add the 2½ cups flour, and teaspoon salt and gently toss to combine. Add the butter and, using a fork or handheld dough blender, cut the butter into the flour, gently stirring to ensure every crumb of butter is pea-sized and coated in flour. Once butter is combined, add in ice water one tablespoon at a time, mixing till dough begins to take shape. Gently knead with your fingers to help bring dough together. If needed add more water a little at a time.
Cut dough into two sections, and gently shape into flat disks. Roll one disk to slightly larger than your pie plate, mold into pie plate and spoon prepared apple mixture into pie plate and spread till filling is even.
Roll second disk of pie dough till slightly larger than pie plate and lay second round of dough on top of filled pie and crimp to seal. (Adventurous bakers: lattice or use cookie cutters to make a decorative top crust for your pie.)
Brush the top of the crust lightly with milk and bake for 45–50 minutes, until apples are soft and crust is lightly golden.
Let cool for 10–15 minutes, and serve with a healthy dollop of crème fraîche or vanilla crème fraîche. (OPTIONAL: Lightly whip crème fraîche and sweeten to taste.)