1. Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
2. In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy.
3. Beat in the egg and vanilla extract, then stir in the pumpkin puree until well combined.
4. In a separate bowl, whisk together the flour, baking soda, cinnamon, nutmeg, cloves, and salt.
5. Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
6. Drop by rounded tablespoons onto the prepared baking sheets.
7. Bake for 10-12 minutes, or until the edges are lightly golden. Let cool on the baking sheets for a few minutes before transferring to a wire rack to cool completely.

For the Cinnamon Frosting:

1. In a large bowl, cream together the softened butter and powdered sugar until smooth.
2. Gradually add the milk, vanilla extract, and cinnamon, mixing until the frosting reaches your desired consistency. Add more milk if needed.
3. Once the cookies are completely cool, frost them with the cinnamon frosting.