For the Cookies:
1. Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
2. In a large bowl, cream together the softened butter and granulated sugar until light and fluffy.
3. Beat in the eggs one at a time, then stir in the vanilla extract and pumpkin puree until well combined.
4. In a separate medium bowl, whisk together the all-purpose flour, baking soda, salt, cinnamon, ginger, and nutmeg.
5. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
6. Drop rounded tablespoons of dough onto the prepared baking sheets, about 2 inches apart.
7. Bake for 10-13 minutes, or until the edges are lightly golden and the centers are set but still soft.
8. Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.

For the Cream Cheese Frosting:
1. In a large bowl, beat the softened cream cheese and softened butter together until smooth and creamy.
2. Gradually add the powdered sugar, beating until well combined and smooth. A hand mixer or stand mixer is recommended.
3. Stir in the vanilla extract.
4. If the frosting is too thick, add milk or cream, one tablespoon at a time, until it reaches a spreadable consistency.
5. Once the cookies are completely cooled, spread a generous amount of frosting over the top of each cookie.