Preheat the oven to 350 degrees F and line a large baking sheet with a silpat mat or parchment paper.: You should feel the warmth building in your kitchen as the oven reaches temperature, and the scent of residual warmth if your oven was recently used. Preheating ensures even baking, and using a silpat mat or parchment paper prevents sticking and promotes even browning. A common mistake is not preheating fully, which causes uneven rise and dense centers, so wait until the oven signals it is at temperature before baking.
In a large bowl, beat the butter, granulated sugar, and brown sugar together until well-mixed. If using a stand mixer, make sure to use the paddle attachment.: As you cream the butter with both sugars, listen for a change from clumpy to smooth and watch the mixture lighten in color. This aeration traps tiny air pockets that help the cookies become tender and slightly risen. Overbeating can make the dough too soft and lead to spreading, so stop when the mixture is homogenous and pale.
With the mixer on low speed, add the egg, pumpkin puree, and vanilla extract.: When you add the egg and pumpkin puree , the batter will loosen and become glossy. The pumpkin puree brings moisture and a dense, silky feel, so incorporate it slowly to keep the emulsion stable. If you add everything too quickly, you may end up with a curdled or separated texture; mix until smooth and scrape the bowl to integrate any pockets of unmixed ingredients.
With the mixer on low speed, slowly add the flour, pumpkin pie spice, and baking soda. Mix just until combined, you don’t want to overmix!: The moment the dry ingredients hit the wet, you will hear a muffled, thicker whir from the mixer and see the dough take on a structured appearance. Mixing briefly preserves tenderness by preventing excess gluten development. Overmixing will produce a tough cookie, so stop when you no longer see streaks of flour.
Scoop out 1-inch balls of dough onto the prepared baking sheet, making sure to place them at least 2 inches apart. Press each of the cookies down gently to flatten them slightly.: The dough should feel moist and supple, not sticky or crumbly, and the scooped balls should hold shape. Pressing them slightly promotes even baking and encourages the characteristic soft but set center. If you space them too close, the cookies will merge into one large cookie, so keep the gap consistent.
Bake until the edges are golden brown and the center is set, about 10-12 minutes.: Watch for the edges to take on a light golden tone while the centers remain slightly pale and springy to the touch. You should hear a quiet crackle if the edges are achieving crispness. Removing them too late will dry them out, and removing them too early will leave raw centers, so aim for that gentle spring in the middle.
Transfer the cookies to a wire rack to cool while you make the icing.: The moment the cookies leave the oven you will smell concentrated warmth and spice. Moving them to a wire rack prevents steaming and sogginess by allowing air to circulate. Avoid stacking hot cookies which traps steam and softens edges; let them cool until they reach room temperature before frosting.
Beat the softened cream cheese on medium speed until smooth, about 30 seconds.: The cream cheese should whip into a silky, pale cream without lumps. A short beat time prevents overheating and melting which would make the icing runny. If the cream cheese is not fully softened, you will get clumps, so let it soften to room temperature before starting.
With the mixer on low, slowly add the powdered sugar, about 1/4 cup at a time.: Adding powdered sugar gradually helps maintain a smooth texture and prevents a cloud of sugar from escaping the bowl. As you incorporate it, the frosting will thicken and take on a satiny sheen. Add too quickly and you risk a gritty texture or a sugar cloud, so be patient and mix until fully integrated.
When the sugar has been fully incorporated, add the vanilla and mix to combine. Scrape down the sides of the bowl as needed.: The vanilla extract brightens the frosting and ties it to the cookie flavors. Scraping ensures no pockets of sugar or cream cheese are left behind, resulting in an even, spreadable icing. If you skip scraping, you may end up with uneven sweetness or texture in the frosting.
Frost each of the cookies with 1 tablespoon of icing and top with a pinch of pumpkin pie spice if desired.: The creamy cream cheese frosting should spread smoothly and hold its shape without sliding off. Taste a dab to check balance, then apply about a tablespoon per cookie for a pleasant proportion. Too much frosting can overpower the cookie, while too little will not provide the intended contrast, so measure roughly to get consistent results.
Store leftover cookies in an airtight container in the fridge for up to two days.: Cooling and refrigerating helps the cream cheese frosting stay fresh and prevents spoilage. Place parchment between layers to avoid sticking. Leaving them at room temperature for extended periods risks the frosting softening or becoming unsafe to eat, so refrigerate if not serving within a few hours.