Preheat the oven to 350 degrees F. Melt the butter in a medium saucepan over medium heat. Whisk constantly for about 4 minutes until the butter has a nutty aroma and brown flecks appear. Remove from heat and pour into a heatproof bowl to cool completely.

In a stand mixer fitted with the paddle attachment, add the cooled brown butter, brown sugar, and granulated sugar. Whisk until smooth. Add the eggs and vanilla extract, mixing until thoroughly combined.

In a separate bowl, whisk together the flour, oats, baking soda, salt, cinnamon, cardamom, nutmeg, ginger, cloves, and allspice. Pour the dry ingredients into the mixer and beat on low until just combined. Let the dough rest at room temperature for 10 minutes.

Line a baking sheet with parchment paper. Scoop heaping tablespoons of dough and roll into balls. Place on the baking sheet. Bake for 10-12 minutes until golden brown. Let cool completely.

To make the frosting, beat the room temperature butter and cream cheese in a stand mixer on medium speed for 2-3 minutes until light and fluffy. Gradually add the powdered sugar, mixing on low. Add the vanilla extract, cinnamon, cardamom, nutmeg, ginger, cloves, allspice, and salt. Beat for 30 seconds, then add the milk and beat on high for 1-2 minutes until fluffy.

Place the cookies bottom side up and pipe a tablespoon of frosting in the center of 12 cookies. Top with the remaining cookie halves.