Preheat oven to 350°F (177°C).
Line two baking sheets with parchment paper.
Add pumpkin puree to a fine mesh strainer. Let sit 10–15 minutes to remove excess moisture.
In a large bowl, whisk together flour, baking powder, baking soda, salt, pumpkin pie spice, cinnamon, and ginger.
In another bowl, cream softened butter, brown sugar, and granulated sugar for 2 minutes.
Add the egg and mix for 1 minute.
Mix in strained pumpkin and milk.
Gradually add dry ingredients to wet mixture on low speed. Mix until just combined — dough will be thick and sticky.
Scoop dough (2 tbsp each) onto lined baking sheets, spacing 2 inches apart.
Bake for 14–15 minutes, until edges are set but centers are soft.
Cool completely on racks.
Melt butter in a saucepan over medium heat. Cook 5–7 minutes until butter browns and smells nutty.
Pour into a heat-safe bowl, including browned bits. Chill 20–30 minutes.
Whip chilled butter for 1 minute.
Add powdered sugar and vanilla.
Add buttermilk 1 tablespoon at a time until smooth.
Whip 1–3 minutes until fluffy.
Spread or pipe buttercream onto the bottom of one cookie.
Top with a second cookie.
Repeat with remaining cookies.