In a bowl, whisk together flour, baking powder, and salt. In another bowl, cream the softened butter and granulated sugar until light and fluffy. Beat in the egg and vanilla extract. Gradually add the dry ingredients to the wet ingredients, mixing until a soft dough forms.
Wrap the dough in plastic wrap and refrigerate for at least 1 hour. This step is crucial for the cookies to hold their shape.
Combine melted butter, brown sugar, and cinnamon in a bowl. Set aside.
On a lightly floured surface, roll out the dough into a rectangle. Spread the filling evenly over the dough, leaving a small margin at the edges.
Carefully roll the dough into a log, starting from the long side. Chill the log for 30 minutes.
Preheat your oven to 350°F (175°C). Cut the log into 1/2-inch slices and place them on a baking sheet lined with parchment paper. Bake for 10-12 minutes or until lightly golden.
While the cookies cool, whisk together powdered sugar, milk, and vanilla extract to create a smooth icing.
Drizzle the icing over the cooled cookies and let it set.