Preheat the oven to 350°F (175°C).
In a large bowl, cream together the softened butter and sugar until light and fluffy.
Add the egg and vanilla extract, mixing until well combined.
In a separate bowl, whisk together the flour, salt, baking powder, and cinnamon.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
In another bowl, combine the diced apples, brown sugar, lemon juice, and nutmeg.
Roll out the cookie dough on a floured surface to about 1/4 inch thick.
Use a cookie cutter to cut out shapes, placing half on a baking sheet.
Place a spoonful of the apple mixture on each cut-out cookie, leaving space around the edges.
Top with another cookie cut-out, sealing the edges with a fork.
Bake for 15-20 minutes until golden brown.
Allow cookies to cool before serving, and dust with powdered sugar if desired.