In a large bowl, whisk together the flour, powdered sugar, baking powder, cinnamon, and salt.
Add the cold butter, and cut it into the dry ingredients using a pastry blender or your fingertips until the mixture resembles coarse crumbs.
Stir in the vanilla sugar, egg, and grated apple. Knead gently until a dough forms. Wrap the dough in plastic wrap and refrigerate for 30 minutes.
Preheat oven to 350°F (180°C) and line a baking sheet with parchment paper.
On a lightly floured surface, roll out the dough to a thickness of about 1/8 inch (3-4 mm). Use a round cookie cutter to cut out circles. Cut a smaller hole in the center of every other circle.
Place the cookie rounds on the prepared baking sheet and bake for 10-12 minutes, or until lightly golden brown.
Let the linzer cookies cool completely on the baking sheet. Spread the plain cookies with apple cinnamon jam or preserves, and top with the cookies that have the holes.
Dust with powdered sugar before serving. Garnish with cinnamon sticks or apple slices, if desired.
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