Heat oven to 350° and line two baking sheets with parchment paper. In a large bowl, whisk together flour, baking powder, and salt.
In a large bowl using a hand mixer (or in the bowl of a stand mixer), beat butter and sugar until fluffy and pale in color. Add egg, milk, and vanilla and beat until combined, then add flour mixture gradually until totally combined. Divide dough in half.
On a floured surface, roll each piece of dough into a 9"-x-10" rectangle, about 1/4" thick.
In small bowl, combine brown sugar and cinnamon. Brush rectangles with melted butter and sprinkle with sugar mixture. Tightly roll up each rectangle into a tight log.
Slice into ½"-thick slices and transfer to prepared baking sheets, spaced 1" apart. Bake until lightly golden, 12 to 14 minutes. Let cool on sheets 5 minutes.
In small bowl, combine powdered sugar with milk and drizzle over cookies.