In a food processor, pulse together flour, salt, and sugar. Cut butter into small chunks and add it to the flour. Pulse until small crumbles form.
Slowly drizzle in the ice water a tablespoon at a time and pulse a few times to form a cohesive dough. Dump dough onto a floured counter top and form into a disk. Wrap in plastic and let rest in the refrigerator while you prepare the rest of the recipe.
Make the filling
Toss the apple slices with the lemon juice. Add the sugar, cornstarch, cinnamon, allspice, and salt, and stir to coat. Set aside.
Make the crumble
In a microwave safe bowl, melt the butter. Add the flour, granulated sugar, brown sugar, and salt, and mix with a fork until combined and crumbs are formed.
Assemble the pie
Roll out the pie crust to fit a rimmed baking sheet and transfer the crust to the pan. If needed, cut off any overhang and fill any gaps.
Scoop the apples into the crust, spreading out to fill the crust evenly. Pour any accumulated juices over the top.
Sprinkle the crumble evenly over the top of the apples. Bake the pie for 40-45 minutes, until golden brown.
Make the caramel
While the pie cools, make the sauce. Combine butter, dark brown sugar, and cream in a small saucepan over medium heat and melt, stirring. Once melted, increase slightly and bring to a full rolling boil for 1 minute.
Remove from heat and whisk in vanilla (be careful, as the caramel will bubble up). Add sifted powdered sugar and whisk until completely incorporated and smooth. Whisk in salt to taste. Drizzle over the pie. (Any remaining sauce can be saved in the refrigerator.)