Preheat the oven to 350 degrees F. In a large bowl, cream together butter and sugars for the cookies. Add the eggs and vanilla and mix well. Stir in the pumpkin.
In a medium bowl, sift together the baking soda, cinnamon, cloves, nutmeg, ginger, salt and flour. Add the dry mixture to the pumpkin batter and mix until combined.
Drop the dough by large tablespoonfuls onto a lightly greased (or lined with parchment or silpat) cookie sheet. Bake for 10-12 minutes.
For the frosting, in a medium saucepan, combine the butter, cream, brown sugar and salt, and cook over medium heat, stirring constantly, until the mixture comes to a boil.
Remove from the heat and stir in 1 cup powdered sugar and the vanilla. Let the frosting cool to room temperature and stir in the remaining 1 cup powdered sugar. Spread the frosting over the cooled cookies.