In a medium bowl, sift together flour, baking powder, and salt. Set aside.
In a mixer bowl fitted with the paddle attachment, beat butter and sugar on medium-low speed for 1 minute. Beat in egg and vanilla extract until combined. Reduce speed to low. Add flour mixture and beat just until combined. Do not overmix.
Form dough into a ball (dough should be soft but if it's hard to handle, add a bit more flour), and cut into 2 equal pieces. Flatten each dough to form a disc and wrap with plastic wrap. Refrigerate for 1-2 hours, until firm enough to roll.
Take one piece of dough out of the fridge and let it sit on the counter for a few minutes to soften slightly for easy rolling. On a lightly floured surface (or between 2 pieces of parchment paper), roll the dough into a 10x8 inch rectangle, to about ¼-inch thickness. Spread the dough with half of the butter (1 tablespoon). Mix 1/4 cup sugar and cinnamon in a bowl, then sprinkle half the mixture on top of the butter and press gently.
Roll up dough to form a log, starting with the longer edge. If dough is too soft to cut at the point (and it probably is), chill log in the fridge until firm, about 30 minutes. You can transfer it to a plate and cover loosely with plastic wrap. Repeat the process with the second dough.
Preheat oven to 350F/180C. Line 2 baking sheets with parchment paper.
Move logs from the fridge. Cut into 1/2 inch slices, and place cut side up on the baking sheet, 1-2 inches apart.
Bake for 10-12 minutes, or until cookies appear golden brown at the edges. The cookies should be bright and soft, so don't be tempted to bake them for longer. Allow cookies to cool in the pan for 5-10 minutes, then gently transfer to a wire rack to cool completely.
Make the icing: Whisk the ingredients together in a bowl, adding little milk at a time (you might not need to use all of it) until you reach a thick but pourable consistency. Then, drizzle over the cookies using a spoon or a piping bag.
Store cookies at room temperature in an airtight container for up to a week or freeze for up to 2 months. To thaw, leave on the counter, still covered, or overnight in the fridge.