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.
4. In a separate medium bowl, whisk together the all-purpose flour, baking soda, salt, cinnamon, ginger, nutmeg, and cloves.
5. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
6. Drop rounded tablespoons of dough onto the prepared baking sheets, about 2 inches apart.
7. Bake for 10-12 minutes, or until the edges are set and the centers are still slightly soft. Do not overbake.
8. Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.

For the Frosting:
1. In a medium bowl, beat the softened cream cheese and softened butter together until smooth and creamy.
2. Stir in the vanilla extract.
3. Gradually add the sifted powdered sugar, mixing until the frosting is smooth and reaches your desired consistency.
4. Once the cookies are completely cool, spread or pipe the cream cheese frosting onto the bottoms of one cookie and top with another, or frost the tops of individual cookies.