In a large mixing bowl, cream together the sugar and butter until light and fluffy. Add in the egg and vanilla extract, and continue to beat until well combined.
In a separate bowl, sift together the flour and baking powder. Add the flour mixture to the butter mixture in several stages, mixing well to combine. Dough will be crumbly.
Press the dough together with your hands and then roll out into two 9"x12" rectangles on parchment paper or a floured surface. Dough should be about 1/4" thick.
Combine the filling's cinnamon and sugar in a small bowl, and melt the needed butter, allowing it to cool.
Cover the rectangle's surface with a coat of melted butter, paying close attention to edges. Sprinkle the cinnamon and sugar mixture over the top of the butter, using fingertips to press it lightly into the dough.
Carefully roll up the dough into a log, wrap in plastic wrap and allow to chill for about 2 hours in the refrigerator.
Preheat oven to 350.
Remove the dough logs from the fridge and cut into coins about 1/4" thick, and place on a baking sheet covered in parchment paper.
Bake for 9-10 minutes, or until edges are just barely golden. Allow several minutes to cool on sheet before removing to a rack to cool completely.
Once cool, whisk together all icing ingredients. Pour the icing into a decorating bag and snip the tip of the bag to create a small hole.
Ice cookies in several lines left to right and then top to bottom, or in the design of your choosing.