In a large mixing bowl, cream together the butter and sugar until light and fluffy. Add the egg and vanilla extract, mixing well. In another bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms. Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.

In a saucepan, combine the diced apples, brown sugar, cinnamon, lemon juice, and flour. Cook over medium heat until the apples are tender and the mixture has thickened. Remove from heat and let cool completely.

Preheat the oven to 350°F (175°C). Roll out the chilled dough on a floured surface to about 1/4 inch thick. Use a cookie cutter to cut out circles from the dough.

Place a spoonful of apple filling in the center of half of the dough circles. Top with another dough circle and press the edges to seal. Crimp the edges with a fork for a decorative touch.

Transfer the assembled cookies to the prepared baking sheet. Bake in the preheated oven for 12-15 minutes, or until the edges are golden brown. Allow to cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.

Serve the Apple Pie Cookies warm or at room temperature. They pair wonderfully with a glass of cold milk or a warm cup of tea. Enjoy the comforting flavors of fall with every bite.

Store the cookies in an airtight container at room temperature for up to 5 days. For longer storage, freeze the cookies in a freezer-safe bag for up to 3 months.