In a large bowl, whisk together the flour, baking powder, powdered sugar, and salt.
Add the cold, cubed butter and cut it into the flour mixture using a pastry blender or your fingertips until the mixture resembles coarse crumbs.
Add the egg and mix until the dough comes together into a smooth ball.
Wrap the dough in plastic wrap and chill in the refrigerator for 30 minutes.
Meanwhile, peel and grate the apples. In a separate bowl, combine the grated apples with the chestnut puree, cinnamon, and vanilla extract.
Divide the chilled dough in half. On a lightly floured surface, roll out each half of the dough into a thin circle.
Place one circle of dough into a baking pan lined with parchment paper. Spread the chestnut and apple filling evenly over the dough.
Carefully place the second circle of dough over the filling and gently press the edges to seal.
Bake in a preheated oven at 350°F (180°C) for 25-30 minutes, or until golden brown.
Let the linzer tart cool completely before slicing and dusting with powdered sugar before serving.
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