Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.

In a medium bowl, whisk together flour, baking soda, cinnamon, ginger, cloves, and salt.

In a large bowl, cream together butter and both sugars until light and fluffy. Beat in egg and vanilla.

Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.

Stir in pumpkin puree until evenly distributed.

Drop by rounded tablespoons onto prepared baking sheets, leaving about 2 inches between cookies.

Bake for 10-12 minutes, or until edges are lightly golden brown.

Let cookies cool on baking sheets for a few minutes before transferring to a wire rack to cool completely.

For the frosting: Beat cream cheese and butter until smooth. Gradually add powdered sugar, beating until light and fluffy. Stir in vanilla and salt.

Once cookies are completely cool, frost generously.