Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.
In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the egg and maple extract.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add to the wet ingredients, mixing until just combined.
Drop by rounded tablespoons onto the prepared baking sheets.
Bake for 8-10 minutes, or until the edges are lightly golden brown. Let cool on baking sheets for a few minutes before transferring to a wire rack to cool completely.
Prepare the icing: In a medium bowl, whisk together the powdered sugar, maple syrup, and milk until smooth. Add more milk if needed to reach desired consistency.
Once the cookies are completely cool, spread the icing over the top. Let the icing set before serving.
Notes
Store cookies in an airtight container at room temperature.