Prepare Dry Ingredients: In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, ginger, nutmeg, and salt. Set aside.
Cream Wet Ingredients: In a large bowl, using an electric mixer, cream the softened butter, brown sugar, and granulated sugar together until the mixture is light and fluffy (about 2-3 minutes).
Add Egg and Vanilla: Beat in the egg and vanilla extract until well combined, scraping down the sides of the bowl as needed.
Mix in Pumpkin Puree: Add the pumpkin puree and mix on low speed just until combined. Be sure to scrape the bowl to ensure the pumpkin is fully incorporated into the batter.
Combine Wet and Dry: Gradually add the whisked dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix the dough.
Chill the Dough: Cover the bowl with plastic wrap and chill the dough in the refrigerator for 30-45 minutes. This step helps prevent the cookies from spreading too much during baking.
Preheat Oven & Prep Baking Sheets: While the dough chills, preheat your oven to 350°F (175°C). Line baking sheets with parchment paper.
Scoop & Bake: Once chilled, use a cookie scoop (or two spoons) to scoop rounded tablespoons of dough onto the prepared baking sheets, spacing them about 2 inches apart. Bake for 10-12 minutes, or until the edges are lightly golden and the centers are set.
Cool Cookies: Let the cookies cool on the baking sheets for about 10 minutes before carefully transferring them to a wire rack to cool completely.
Make the Glaze: While the cookies are cooling, prepare the glaze. In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Add more milk, a tiny bit at a time, if you prefer a thinner glaze.
Glaze Cookies: Once the cookies are mostly cooled, drizzle the vanilla glaze generously over the top of each cookie. Allow the glaze to set before serving.