In a 2-quart sauté pan over medium heat, melt the butter.
Add the apples and sauté, stirring occasionally, until the apples are tender and have released most of their liquid, 5 to 7 minutes.
Add the sugar and cardamom, stir to dissolve and cook, stirring occasionally, until most of the liquid has evaporated, 12 to 15 minutes more.
Remove from the heat and let the apple compote cool to room temperature.
Position a rack in the center of an oven and preheat to 375 degrees F.
Line a baking sheet with parchment paper.
With a sharp knife, cut the cheese in half horizontally.
On a clean work surface, place one half of the cheese, sliced side up, and evenly spread 1/2 cup of the apple compote over it. Set the other half, sliced side down, over the compote and spread an additional 1/2 cup compote over the top.
Place the dough on a lightly floured surface and set the cheese in the center of the dough.
Fold the dough up over the sides of the cheese, pleating the upper edges to fit snugly around the cheese.
Pinch the dough together in the center to seal.
Brush the dough evenly with the egg wash and place on the prepared baking sheet.
Bake until the pastry is golden all over and crisp, 40 to 45 minutes.
Let it rest for 5 minutes, then transfer to a platter along with a sharp knife and the crackers.