In a medium bowl, whisk together the flour, baking powder, baking soda, kosher salt, cinnamon, ginger, and nutmeg. Mix thoroughly so the leavening agents and spices are evenly distributed. Set the bowl aside.

In a large mixing bowl, beat the softened butter, granulated sugar, and brown sugar with an electric mixer on medium speed for 2–3 minutes. The mixture should look light, creamy, and evenly combined. Scrape down the sides of the bowl as necessary.

Beat in the egg and vanilla extract until incorporated. Add the pumpkin puree and mix on low speed just until combined. The mixture may look slightly separated because pumpkin contains considerable moisture. This is normal and will resolve when the dry ingredients are added.

Add the dry ingredients to the pumpkin mixture in two additions. Mix on low speed or fold with a rubber spatula until no dry flour remains. The dough should be thick, soft, and sticky, similar to a very thick cake batter. Avoid overmixing, which can make the finished cookies tough.

Cover the bowl and refrigerate the dough for 30 minutes. The dough should become noticeably firmer and easier to scoop. Chilling helps the cookies bake into rounded mounds rather than spreading into thin disks.

Near the end of the chilling time, preheat the oven to 350°F (175°C). Line two large baking sheets with parchment paper. Position the oven racks in the upper and lower thirds of the oven.

Use a 1½-tablespoon cookie scoop to divide the chilled dough into 24 portions, placing 12 cookies on each prepared baking sheet. Leave approximately 2 inches between the portions. The dough will be sticky, so there is no need to roll it by hand. Gently smooth any tall peaks with the back of a lightly dampened spoon if desired.

Bake for 11–13 minutes, switching the baking sheets between racks and rotating them halfway through. The cookies should look puffed with set edges and matte, lightly springy tops. They will remain fairly pale, so do not wait for deep browning. If a cookie's center still looks wet, bake for another 1–2 minutes.

Allow the cookies to rest on their baking sheets for 5 minutes, then transfer them to a wire cooling rack. Let them cool completely for approximately 40 more minutes before frosting. The cookies should feel soft but firm enough to lift without breaking. Frosting them while warm will cause the topping to melt.

In a medium bowl, beat the softened cream cheese and butter together for approximately 2 minutes, until smooth and creamy. Add the sifted powdered sugar, cinnamon, and vanilla extract. Beat on low speed until incorporated, then increase to medium speed and mix until fluffy. The frosting should be thick enough to hold soft swirls. If it is too stiff to spread, mix in the milk 1 teaspoon at a time, using no more than 2 teaspoons.

Spread approximately 1 tablespoon of cinnamon frosting over the top of each cooled cookie using an offset spatula or butter knife. Create gentle swirls across the surface, leaving a small border around each cookie if desired. For a neater finish, refrigerate the frosting for 10 minutes before spreading if it becomes too soft.

Lightly dust the frosted cookies with the optional ground cinnamon. Serve immediately or refrigerate until the frosting firms up. For the softest texture, allow chilled cookies to stand at room temperature for 10–15 minutes before serving.