Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.

In a medium skillet over medium heat, melt 2 tablespoons of butter. Add the diced apples and cook until slightly softened, about 5-7 minutes.

Stir in the brown sugar, cinnamon, and lemon juice. Continue to cook for another 3-5 minutes, or until the apples are tender and the liquid has slightly thickened. Remove from heat and let cool completely.

Unroll the thawed puff pastry sheet onto a lightly floured surface. Cut the pastry into four equal squares.

Spoon a generous amount of the cooled apple filling onto one half of each pastry square, leaving a small border.

Fold the other half of the pastry over the filling to create a triangle. Crimp the edges with a fork to seal them tightly. Cut a few small slits on top of each turnover to allow steam to escape.

In a small bowl, whisk the beaten egg with 1 tablespoon of water to create an egg wash. Brush the tops of the turnovers with the egg wash.

Place the turnovers on the prepared baking sheet and bake for 20-25 minutes, or until golden brown and puffed. Let cool slightly on a wire rack.

While the turnovers are baking, prepare the miso-caramel sauce. In a small saucepan, combine the granulated sugar and 2 tablespoons of water. Cook over medium heat, without stirring, until the sugar melts and turns into a deep amber caramel. Swirl the pan occasionally.

Carefully and slowly pour in the heavy cream (it will bubble vigorously). Stir until smooth. Add the butter, white miso paste, vanilla extract, and a pinch of salt. Whisk until the butter and miso are fully incorporated and the sauce is smooth. Remove from heat.

Once the turnovers have cooled slightly, drizzle generously with the warm miso-caramel sauce. Serve immediately and enjoy!