In a large bowl, beat the softened butter and granulated sugar until smooth and creamy, about 2 minutes. Add the egg and vanilla extract, beating until fully combined.
Mix together the flour, baking powder, and salt in a medium bowl.
Gradually add the dry ingredients to the wet ingredients, mixing until a soft dough forms.
Divide the dough in half. On a floured surface, roll each half into a 9x7 inch rectangle. Spread melted butter over each rectangle.
Mix cinnamon sugar ingredients together, then sprinkle the mixture on top of the melted butter. Roll up each rectangle tightly, wrap in plastic, and chill in the refrigerator for 2 hours.
Preheat the oven to 350°F. Cut the chilled dough into 1/2 inch slices and place on a baking sheet lined with parchment paper. Bake for 10 minutes or until lightly golden.
Mix powdered sugar and milk to create a smooth icing, starting with less milk and adding more if needed to reach your desired consistency. Drizzle over cooled cookies.