In a large bowl, cream the softened butter and sugar until light and fluffy (about 2 minutes). Beat in the egg and vanilla extract. In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet, mixing until a soft dough forms.

Divide the dough in half. Roll one portion between two sheets of parchment paper into a rectangle about ¼ inch thick. Brush with melted butter, then sprinkle evenly with the brown sugar and cinnamon mixture. Roll the dough tightly into a log, wrap in plastic wrap, and chill for at least 1 hour. Repeat with the second half of dough.

Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper. Slice the chilled dough logs into ¼ to ½ inch thick rounds and place them 2 inches apart on the sheets. Bake for 10–12 minutes, or until the edges are lightly golden. Let cool on the sheet for 5 minutes before transferring to a wire rack.

While cookies are cooling, whisk together the powdered sugar, milk, and vanilla extract until smooth. Drizzle the glaze over cooled cookies and allow it to set before serving.