Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, cream butter, brown sugar, and granulated sugar until fluffy.
Add pumpkin puree, egg, and vanilla. Mix until well combined.
In a separate bowl, whisk together flour, baking soda, baking powder, salt, and spices.
Gradually add dry ingredients to wet ingredients, mixing until just combined.
Drop dough by rounded tablespoons onto prepared baking sheets.
Bake for 12-14 minutes, or until edges are set and centers are slightly soft.
Let cookies cool on the baking sheet for 5 minutes, then transfer to wire racks to cool completely.
For frosting, beat mascarpone and butter until smooth.
Add powdered sugar, vanilla, and salt. Beat until light and fluffy.
Spread frosting generously over cooled cookies.
Store in an airtight container in the fridge.