In a large bowl, whisk together flour, sugar, and salt. Cut in cold butter with a pastry blender until mixture resembles coarse crumbs with pea-sized butter pieces.
Sprinkle in ice water one tablespoon at a time, tossing with a fork, until dough just holds together. Divide into two equal disks, wrap in plastic, and refrigerate for at least one hour.
In a medium saucepan over medium heat, melt sugar undisturbed until it turns deep amber. Whisk in butter, then slowly stream in heavy cream while whisking constantly. Remove from heat and stir in sea salt. Cool to room temperature.
In a large bowl, toss sliced apples with sugars, flour, lemon juice, cinnamon, nutmeg, and salt. Pour half the cooled caramel over the apples and gently fold to coat, reserving the rest for topping.
On a floured surface, roll one dough disk into a 12-inch circle. Transfer to a 9-inch pie dish, gently press into bottom and sides, and trim excess dough to 1-inch overhang.
Pour the caramel-coated apple filling into the prepared crust, mounding slightly in the center. Discard any excess liquid left in the bowl.
Roll the second dough disk into a 12-inch circle and cut into strips. Weave a lattice over the filling, or place the whole round on top with slits. Trim and crimp edges to seal.
Preheat oven to 375°F. Brush top crust with egg wash and sprinkle with coarse sugar. Place a pie crust shield over edges and bake on a baking sheet for 55-65 minutes, removing shield for last 15 minutes.
Cool pie on a wire rack for at least 3 hours. Warm reserved caramel and drizzle over the top just before serving. Finish with an extra pinch of flaky sea salt.