Prepare the Pie Crust
In a large bowl, whisk together the flour, sugar, and salt. Add the cold cubed butter and use a pastry cutter or your fingertips to work it into the flour until the mixture resembles coarse crumbs with some pea-sized pieces remaining.Gradually add ice water, one tablespoon at a time, mixing gently until the dough just comes together. Divide the dough into two equal portions, form into disks, wrap in plastic wrap, and refrigerate for at least 1 hour.
Make the Miso Caramel
In a medium saucepan, combine the sugar and water, stirring until the sugar is moistened. Heat over medium-high heat without stirring until the mixture turns an amber color, about 8-10 minutes.Carefully whisk in the room temperature heavy cream (it will bubble vigorously). Add the butter and whisk until melted and smooth. Whisk in the miso paste and salt until fully incorporated. Transfer to a heat-proof container and let cool.
Prepare the Apple Filling
Peel, core, and thinly slice the apples. In a large bowl, toss the sliced apples with lemon juice. In a separate bowl, combine the sugar, cornstarch, cinnamon, nutmeg, allspice, and salt. Sprinkle this mixture over the apples and toss until evenly coated.Assemble the Pie
Preheat your oven to 375°F (190°C). Roll out one disk of dough on a floured surface to a 12-inch circle, about ⅛-inch thick. Carefully transfer to a 9-inch pie dish.Pour ⅓ of the miso caramel into the bottom of the pie crust. Layer in the apple filling, mounding slightly in the center. Drizzle another ⅓ of the miso caramel over the apples, reserving the final ⅓ for serving.
Roll out the second disk of dough and either place it whole over the filling with slits cut for venting, or create a lattice top. Trim excess dough and crimp the edges to seal.
Bake the Pie
Brush the top crust with egg wash and sprinkle with turbinado sugar. Place the pie on a baking sheet and bake in the preheated oven for 20 minutes. Reduce the temperature to 350°F (175°C) and continue baking for 35-40 minutes more, until the crust is golden brown and the filling is bubbling.Cool and Serve
Allow the pie to cool on a wire rack for at least 2 hours before serving. Serve warm or at room temperature with the reserved miso caramel drizzled on top.