Preheat your oven to 350°F (175°C). Line two large baking sheets with parchment paper or silicone baking mats. Position the oven rack in the center for even baking.

In a large mixing bowl, whisk together the pumpkin puree and melted, slightly cooled butter until smooth. Add the granulated sugar and brown sugar and whisk until thick and glossy. Whisk in the egg and 2 teaspoons vanilla extract until the mixture is smooth and creamy.

In a separate bowl, whisk together the flour, baking soda, baking powder, ground cinnamon, pumpkin pie spice, and fine sea salt until well combined and no clumps remain.

Add the dry ingredients to the wet ingredients in two additions, stirring gently with a spatula or wooden spoon after each addition. Mix just until no streaks of flour remain. The dough will be thick, soft, and slightly sticky—similar to a soft cake batter. Avoid overmixing so the cookies stay tender.

Using a medium cookie scoop (about 1 1/2 tablespoons of dough), portion mounds of dough onto the prepared baking sheets, spacing them about 2 inches apart. Lightly dampen your fingertips and gently smooth the tops if needed. Bake one sheet at a time for 11–13 minutes, or until the edges are set, the tops look dry and slightly puffed, and the centers spring back lightly when touched. The cookies will continue to set as they cool.

Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely before frosting.

In a medium bowl, beat the softened cream cheese and 1/4 cup softened butter together with a hand mixer or stand mixer on medium speed until smooth and fluffy, about 2–3 minutes. Add 1 teaspoon vanilla extract and a pinch of salt and mix again. Gradually add 2 cups of sifted powdered sugar, mixing on low at first, then on medium until smooth and creamy. If the frosting is too thin, add more powdered sugar a couple of tablespoons at a time. If it is too thick, mix in 1–2 teaspoons of heavy cream or milk at a time until it reaches a spreadable consistency.

Once the cookies are completely cool, generously spread cream cheese frosting over the tops with a small spatula or butter knife. If desired, lightly sprinkle cinnamon or pumpkin pie spice over the frosted cookies. Let them sit at room temperature for 20–30 minutes so the frosting can set slightly, especially if you plan to stack or pack them.

Store frosted cookies in an airtight container in the refrigerator for 4–5 days. For the best texture, let them sit at room temperature for 15–20 minutes before serving so the frosting softens slightly. Unfrosted cookies can be frozen for up to 2 months in an airtight container or freezer bag, then thawed at room temperature and frosted. You can also freeze frosted cookies, well wrapped, though the frosting may show slight freezer marks.