In a medium-sized saucepan, melt the butter over medium heat. Stir continuously until it foams and the milk deposits separate at the bottom. Continue stirring until the butter turns brown and crackling noises are heard. Remove from heat.
Pour the browned butter into a glass bowl and allow it to cool completely to room temperature. While it's cooling, you can prepare the cookie dough.
For the Cookies:
Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper. Set it aside.
In a stand mixer fitted with the paddle attachment, cream the cold, cubed butter and brown sugar on medium-high speed until the mixture becomes light and fluffy (about 2-3 minutes). Add the pumpkin puree and mix just until combined.
Add the egg and vanilla, and beat until the mixture is smooth (about 30 seconds).
Add the flour, baking powder, pumpkin pie spice, cinnamon, and salt, and mix on low speed just until combined. Avoid overmixing.
Use a large cookie scoop (I used a 3 tablespoon scoop) to portion out 12-14 equally sized cookie dough balls and place them on the prepared baking sheet.
Lightly spray the tops of the cookie dough balls and flatten them slightly to about ½ inch in thickness using a flat surface, such as a metal spatula.
Bake for 15-16 minutes, then allow the cookies to cool on the baking sheet before transferring them to a cooling rack.
Continuing with Cream Cheese Frosting:
In a bowl of a standard mixer fitted with the paddle attachment, combine the room temperature browned butter and cream cheese. Beat on medium-high speed for 2-4 minutes until the mixture becomes light and fluffy.
After beating the butter and cream cheese, add the powdered sugar, vanilla, cream, and salt. Mix on low speed until just combined, then increase the speed to high and beat for at least 2 minutes.
Use a rubber spatula or wooden spoon to eliminate any air pockets in the frosting.
Assembling the Cookies:
Once the cookies have completely cooled and are no longer warm, frost the tops.
Using a plastic bag or a piping bag, cut a medium-sized hole and frost the tops of the cookies.
Dust the frosted cookies with ground cinnamon for the finishing touch.