In a medium bowl, whisk together gluten free all-purpose flour blend, baking powder, and salt. Set aside.
In a large bowl, beat softened butter and granulated sugar with an electric mixer until light and fluffy, about 3 minutes. Beat in the egg and vanilla extract until combined. Add the flour mixture and mix on low speed until a smooth dough forms.
Roll the dough out between two sheets of parchment paper into a large rectangle, roughly 10x14 inches and about 1/4 inch thick. In a small bowl, mix the brown sugar and cinnamon together, then sprinkle it evenly over the entire surface of the dough.
Starting from one long edge, tightly roll the dough into a log, using the parchment to help guide it. Wrap the log tightly in plastic wrap and refrigerate for at least 2 hours until very firm.
Preheat your oven to 350 degrees F and line two baking sheets with parchment paper. Slice the chilled log into rounds about 1/3 inch thick. Place them 2 inches apart on the prepared baking sheets.
Bake for 11 to 13 minutes until the edges are lightly golden but the centers still look soft. Let them cool on the baking sheet for 5 minutes, then transfer to a wire rack.
Whisk together powdered sugar, milk, and vanilla extract until a smooth glaze forms. Drizzle generously over the slightly warm cookies. Let the glaze set for 10 minutes before serving.