Brown the butter: Melt the butter in a small saucepan over medium heat. Once melted, whisk constantly; the butter will begin to crackle, then foam. After a few minutes, the butter will begin to turn a golden amber color. As soon as the butter turns an amber brown and gives off a nutty aroma, 5 minutes total, remove from the heat and transfer to the bowl of the electric mixer to allow it to cool for 5 to 10 minutes. Make sure you scrape all the yummy bits of brown butter from the pan into the bowl – that’s where the flavor is! You should end up with roughly 90g to 95g of brown butter.
Squeeze out pumpkin moisture: **While the brown butter cools, measure out ½ cup (120g) pumpkin puree. Place the pumpkin puree in paper towels (I suggest using a few layers) or a clean dish towel and squeeze out ALL the excess moisture, discarding the water. You will be left with approximately 70g to 75g of pumpkin puree (please measure this!).
Combine wet ingredients: Add the pumpkin puree and brown sugar to the brown butter. Using the paddle attachment, mix together on medium speed until well combined, 1 minute. Next, add in the egg and vanilla and mix again on medium speed until smooth and well combined and mixture resembles caramel, about 1 minute.
Mix together dry ingredients: In a separate large bowl, add the dry ingredients: flour, oats, cinnamon, nutmeg, ginger, allspice, baking soda, and salt. Whisk together until combined.
Add the dry ingredients to the wet ingredients and mix on medium-low speed until just combined. Cover the dough and place in the fridge for 30 minutes while you preheat your oven.
Preheat the oven and rest the dough: While the dough chills, preheat the oven to 350 degrees F and line a large baking sheet with parchment paper.
Bake the cookies: After the dough rests, use your hands or a tablespoon to grab 1 ½ tablespoons of dough and roll into a ball. Place dough balls 2 inches apart on the prepared pan and use your fingers or palm of your hand to flatten each ball to be ½ inch thick. Bake for 10 to 12 minutes. If you like cookies with crispy edges, bake for about 12 minutes. If you like softer cookies, bake for 10 minutes. Remove from the oven and cool the cookies on sheets for about 5 minutes then transfer to a wire rack to cool completely. They will still remain soft on the inside. Repeat with remaining dough. Makes 20 cookies (or 10 cookie sandwiches).
Brown the butter for the frosting: While the cookies bake and cool, you can make your brown butter frosting: Melt the 4 tablespoons of butter in a saucepan (you can use the same pan you used initially) over medium heat. Once melted, whisk constantly; the butter will begin to crackle, then foam. After a few minutes, the butter will begin to turn a golden amber color. As soon as the butter turns brown and gives off a nutty aroma, 5 minutes total, pour the brown butter into a small bowl and transfer to the fridge for 30 to 45 minutes to allow the butter to resolidify so that it’s similar to room temperature butter but not completely hardened (poking it should leave an indent). If it hardens too much, allow it to soften up at room temperature.
Make the frosting: In the bowl of a stand mixer fitted with the whisk attachment, beat the cooled brown butter on high speed until light and fluffy, about 1 minute. Add the cream cheese and beat on high until creamy, about 1 minute. Add the powdered sugar, vanilla, and salt and beat until light and fluffy, 2 to 3 minutes.
Assemble the cookies: Once cookies are cool, you can add the filling: Top one cookie sandwich with a spoonful of filling, leaving a nice edge around the cookie and top with another cookie sandwich, then very gently press down slightly. Repeat with all cookies. You should get about 10 to 12 oatmeal cream pies. Feel free to double the recipe if you’d like!