Prepare the Dough: In a mixing bowl, cream the butter and sugar together until light and fluffy. Add the egg and vanilla extract, mixing until combined.
Add Dry Ingredients: Gradually mix in the flour, baking powder, and salt until a soft dough forms. If the dough is sticky, chill for 30 minutes.
Roll Out the Dough: Divide the dough into two equal portions. Roll each portion into a 9x7-inch rectangle on a lightly floured surface or silicone baking mat.
Make the Filling: In a small bowl, mix the softened butter, brown sugar, and cinnamon into a smooth paste. Spread evenly over the rolled-out dough.
Roll into Logs: Carefully roll each rectangle into a tight log, smoothing out any cracks. Wrap in plastic wrap and refrigerate for at least 2 hours.
Slice and Bake: Preheat oven to 350°F (175°C). Slice each log into ½-inch thick rounds and place them on a parchment-lined baking sheet.
Bake the Cookies: Bake for 10-12 minutes or until the edges are lightly golden. Let them cool on the baking sheet for 5 minutes before transferring to a wire rack.
Prepare the Icing: In a small bowl, whisk together powdered sugar, milk, and vanilla extract until smooth.
Drizzle and Serve: Once the cookies are completely cool, drizzle the icing over the top. Allow the icing to set before serving.