In a large mixing bowl, combine all-purpose flour, salt, and granulated sugar. Using a pastry cutter or your fingers, work in chilled unsalted butter until the mixture resembles coarse crumbs. Gradually add ice water, mixing until a dough forms. Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.

In a medium bowl, whisk together brown sugar, granulated sugar, cinnamon, salt, and flour. Cut in cold unsalted butter until the mixture is crumbly, resembling wet sand. Chill the crumb topping in the refrigerator while you prepare the apple filling.

In two separate skillets, over medium heat, melt granulated sugar until it turns a golden caramel, about 5 minutes. Add a portion of the sliced apples to each pan, stirring gently to coat. Cook the apples for 3-4 minutes until just tender but still firm.

In a large bowl, combine the cooled apples, brown sugar, dark rum, cornstarch, vanilla extract, ground cinnamon, nutmeg, and kosher salt. Toss everything together until the apples are well-coated.

Preheat your oven to 425°F (220°C). Roll out the chilled dough on a floured surface, then fit it into a deep-dish pie pan. Pour the apple filling into the crust and evenly distribute it. Sprinkle the crumb topping over the apples, ensuring good coverage. Bake the pie uncovered for 15 minutes, then reduce the temperature to 350°F (175°C) and cover with foil. Continue baking for an additional 30 minutes.

In a small saucepan over medium heat, combine light corn syrup, sugar, unsalted butter, and salt. Bring to a simmer, stirring until the mixture thickens, about 3-5 minutes. Remove from heat and stir in heavy cream and spiced rum.

Once baked, remove the Hot Buttered Rum Apple Pie from the oven and let it cool slightly. Drizzle half of the prepared hot buttered rum sauce over the pie. Reserve the remaining sauce for serving.