In a medium saucepan, melt the butter over medium heat, swirling the pan occasionally. Continue cooking until the butter turns golden brown and develops a nutty aroma, about 5-7 minutes. Be careful not to burn it! Once browned, remove from heat and let it cool slightly. The butter should be warm but not hot. Let it cool for about 10-15 minutes.
While the butter is cooling, Start by blotting your pumpkin. Spread the pumpkin puree evenly on a flat surface, like a plate or a sheet pan. Place a paper towel over the top and lightly press to absorb the excess liquid. Repeat this step at least five more times. You should have about 3-4 tablespoon of pumpkin left. Alternatively, or the cheesecloth method: Place the pumpkin puree in the center of a cheesecloth, twist it tightly to squeeze out the excess liquid, and let it sit for a few minutes before unwrapping and using.
In a large mixing bowl, whisk together the slightly cooled browned butter, granulated sugar, and brown sugar vigorously for 1 ½ minutes. Add the egg and whisk vigorously for 45 more seconds. The mixture should lighten up in color and look like a thick paste. Stir in the vanilla extract and blotted pumpkin puree, and whisk for about 1 minute until fully combined.
Add the flour, baking powder, salt, corn starch, cinnamon, and pumpkin pie spice. Switch to a rubber spatula and gently fold the mixture until just combined.
Scoop the dough into balls using about 3 tablespoons of dough per cookie, Place the dough balls on the prepared baking sheets, spacing them about 2 inches apart. Place in the fridge while the oven preheats.
Preheat the oven to 350°F (177°C) and line two baking sheets with parchment paper. In a medium shallow bowl, mix the ¼ cup granulated sugar, pumpkin pie spice, and optional ground cardamom. Set aside.
When the oven is up to temperature, roll the cookies in the cinnamon-sugar mixture and bake for 10-12 minutes, until the edges are lightly golden and the centers are soft. If the cookies don’t spread flat, gently press them down with the bottom of a cup or spatula to create an even, flat surface. Aim for them to be ¼ inch thick.
Allow the cookies to cool on the baking sheet. While the cookies are cooling, make the frosting.
For the frosting
In the bowl of an electric mixer fitted with the paddle attachment (or using an electric hand mixer), beat the softened butter and cream cheese on medium speed for about 2 minutes. Scrape down the sides and bottom of the bowl, then beat for another 2 minutes. With the mixer on low speed, gradually add the powdered sugar, followed by the vanilla, ground cinnamon, and salt. Once combined, increase the speed to medium and beat until smooth and fluffy, about 1-2 minutes.
Once the cookies are completely cooled, spread or pipe the cream cheese frosting onto the bottom sides of half of the cookies. If using a piping bag, Wilton tip #1M is recommended for even distribution. Sandwich the remaining unfrosted cookies on top, pressing gently to secure.