Preheat oven to 350°F (177°C). Line two baking sheets with parchment paper and set aside.

Place the pure pumpkin into a fine mesh strainer set over a bowl. Allow it to drain for at least 10–15 minutes to remove excess liquid.

In a large bowl, whisk together the flour, baking powder, baking soda, salt, pumpkin pie spice, cinnamon, and ginger. Set aside.

In a separate large bowl, use a hand mixer on medium speed to cream the softened butter, brown sugar, and granulated sugar until smooth, about 2 minutes.

Add the egg and beat on high speed until combined, about 1 minute, scraping down the sides of the bowl as needed.

Add the milk and strained pumpkin, then mix on high speed until combined. The mixture may look slightly lumpy—this is normal.

Add the dry ingredients to the wet ingredients, mixing on low speed until just combined. The dough will be thick and sticky. Using a 2-tablespoon scoop, drop the dough onto the lined baking sheets, about 2 inches apart.

Bake for 14–15 minutes, or until the edges appear set but the center is still soft.

Remove from the oven and let cookies cool completely.

In a medium saucepan, melt the butter over medium heat. Continue cooking, stirring frequently, until the butter foams and brown bits form at the bottom. This takes about 5-7 minutes. Watch closely to avoid burning. When the butter reaches a warm amber color and has a nutty aroma, remove it from heat.

Pour the browned butter into a heat-safe bowl and chill it in the refrigerator for 20-30 minutes.

Once the browned butter is cooled, place it in a stand mixer fitted with the whisk attachment. Whip the butter on medium-high speed for about 1 minute.

Gradually add the confectioners' sugar. Then, add the vanilla.

Reduce the mixer speed to low. Add buttermilk, one tablespoon at a time, until the frosting reaches a smooth and spreadable consistency. Whip the frosting on high speed for 1–3 minutes, until light and fluffy.

Once the cookies are completely cooled, spread or pipe a generous amount of brown butter buttercream on the bottom of one cookie. Top with a second cookie to form a sandwich. Repeat with remaining cookies and buttercream.