In a mixer bowl fitted with the paddle attachment, beat together butter, brown sugar, and granulated sugar on medium speed for 3 minutes. Add pumpkin puree, egg, and vanilla extract, and beat until combined. Add flour mixture and beat on low speed just until combined.
Shape dough into balls, the size of 1.5 tablespoons (30g) each (I like to use this medium scoop), and place on prepared baking sheets, spacing them 2-inches apart. The cookies won't spread much during baking.
Bake for 15-18 minutes, until the edges appear set. Allow cookies to cool for 10-15 minutes in the cookie sheet, then gently transfer them to a wire rack to cool completely.
To make the glaze (optional): Whisk all the ingredients in a small bowl until smooth. Add more maple syrup as necessary until you get a thick yet pourable consistency. Drizzle over cookies using a spoon or a piping bag (it's easier with a piping bag). Allow glaze to set, 15-30 minutes.