Preheat your oven to 350°F (175°C). In a large mixing bowl, cream together the unsalted butter, brown sugar, and granulated sugar for about 3 minutes until light and fluffy. Add pumpkin puree and vanilla, mixing well. In another bowl, whisk together all-purpose flour, baking soda, baking powder, salt, cinnamon, nutmeg, cloves, and pumpkin pie spice. Gradually fold this mixture into the wet ingredients until a sticky dough forms.

Using a tablespoon or cookie scoop, portion out the dough into 2-tablespoon-sized balls on a parchment-lined baking sheet, spacing them about 2 inches apart. Gently flatten each ball slightly. Bake for 15 minutes until edges are lightly golden and centers are still soft. Cool on the baking sheet for a few minutes before transferring to a wire rack.

While cookies cool, prepare the cream cheese frosting. In a mixing bowl, combine room-temperature cream cheese and unsalted butter. Beat on medium speed until smooth and creamy, about 2 minutes. Gradually add powdered sugar and milk, mixing until fluffy and spreadable. Add a splash of vanilla.

Once cookies are completely cooled, frost each with a generous layer of cream cheese frosting. Optionally, sprinkle a pinch of cinnamon or pumpkin pie spice on top.