Preheat the oven to 350 degrees F and line two baking sheets with parchment paper. Whisk together the gluten free flour blend, pumpkin pie spice, baking soda, baking powder, and fine sea salt in a medium bowl. Set aside.
In a large bowl, beat the softened butter and light brown sugar with an electric mixer on medium-high speed until fluffy, about 3 minutes. Add the pumpkin puree, egg, and vanilla extract and beat until smooth and well combined.
Add the flour mixture to the wet ingredients and mix on low speed until just combined into a thick, soft batter. It will be softer than typical cookie dough, which is expected.
Drop the batter by rounded tablespoons onto the prepared baking sheets, spacing them about 2 inches apart. These cookies hold their shape well and do not spread much.
Bake for 13 to 15 minutes until the tops spring back gently when lightly pressed and the edges are just set. Let the cookies cool completely on the baking sheet.
Whisk together the powdered sugar, milk, ground cinnamon, and vanilla extract until smooth and thick but drizzleable. Spoon the glaze over each cooled cookie, spreading it gently with the back of the spoon.
Let the glaze set for about 20 minutes until firm. These cookies are incredibly soft, cake-like, and bursting with warm pumpkin spice flavor.