In a medium bowl, whisk together the all-purpose flour, baking soda, salt, pumpkin pie spice, cinnamon, and nutmeg. Set aside.

In a large bowl, using an electric mixer, cream together the softened butter, granulated sugar, and light brown sugar until light and fluffy, about 2-3 minutes.

Beat in the large egg and vanilla extract until just combined. Scrape down the sides of the bowl. Add the pumpkin puree and mix on low speed until fully incorporated.

Gradually add the dry ingredient mixture to the wet ingredients, mixing on low speed until just combined and no streaks of flour remain. Be careful not to overmix.

Cover the bowl with plastic wrap and chill the dough in the refrigerator for at least 30 minutes. This helps prevent spreading and results in a better cookie texture.

Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper.

Scoop rounded tablespoons of dough onto the prepared baking sheets, spacing them about 2 inches apart. Bake for 10-12 minutes, or until the edges are set and lightly golden, but the centers still look soft. Do not overbake to ensure soft cookies.

Let cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. Cookies must be completely cool before frosting.

While cookies cool, in a medium bowl, beat softened cream cheese and butter together with an electric mixer until smooth and creamy.

Gradually add the sifted powdered sugar, beating until smooth. Add vanilla extract and mix well.

If the frosting is too thick, add milk or cream one tablespoon at a time until desired spreading consistency is reached.

Once cookies are completely cool, frost them generously with the cream cheese frosting. Serve immediately or store.