1. In a food processor, pulse the flour and salt until combined. Add the chilled butter and pulse until the mixture resembles coarse crumbs.

2. Add the ice water a tablespoon at a time, pulsing until the dough just begins to come together.

3. Turn the dough out onto a lightly floured surface and gather into a disk. Wrap in plastic and refrigerate for at least 1 hour.

4. In a small saucepan, combine the granulated sugar and brown sugar. Cook over medium heat, stirring occasionally, until the sugar turns a deep amber color, about 5-7 minutes.

5. Carefully add the heavy cream and miso paste (the mixture will bubble vigorously). Whisk until the miso is fully incorporated and the caramel is smooth. Set aside to cool.

6. Preheat the oven to 375°F (190°C).

7. On a lightly floured surface, roll the chilled dough into a 12-inch circle, about 1/4-inch thick. Transfer to a parchment-lined baking sheet.

8. Arrange the apple slices in a circular pattern on the dough, leaving a 2-inch border. Fold the dough over the apples, pleating it as you go.

9. Brush the dough with the beaten egg.

10. Drizzle the miso caramel sauce over the apples, allowing it to drip down the sides.

11. Bake for 35-40 minutes, until the crust is golden brown and the apples are tender.

12. Allow to cool for 10 minutes before serving.