In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large bowl, cream the butter, granulated sugar, and powdered sugar until light and fluffy. Beat in the egg, milk, and vanilla extract until well combined. Gradually add the dry ingredients and mix until a soft dough forms.
Split the dough in half. Flatten each half into a disc, wrap in plastic wrap, and refrigerate for at least 1 hour.
On a lightly floured surface, roll one portion of chilled dough into a rectangle about ¼ inch thick. Brush with half of the melted butter, then sprinkle evenly with half of the brown sugar and cinnamon mixture.
Starting from the long edge, roll the dough into a tight log. Wrap in plastic and refrigerate again for 30 minutes to firm up for cleaner slicing. Repeat with the second portion of dough.
Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.
Remove dough logs from the refrigerator and slice into ½-inch thick rounds. Place on prepared baking sheets about 2 inches apart. Bake for 10–12 minutes, or until the edges are set and just starting to turn golden.
Cool cookies on the baking sheet for 5 minutes before transferring to a wire rack to cool completely
Whisk together powdered sugar, milk, and vanilla extract until smooth. Drizzle over cooled cookies and allow icing to set before serving.