In a large bowl, start by beating together softened butter and sugar until smooth.
Next, add in the egg, vanilla, pumpkin, and salt and beat again until just combined.
Sift in all dry ingredients and blend everything until well combined.
Using a cookie scoop, scoop rounded dough onto lightly greased cookie sheet. Smooth tops of cookies a bit.
Bake in preheated oven at 350° F for about 15 minutes or until lightly browned on the bottoms. Cool completely on a wire rack before frosting. Makes about 3 dozen cookies.
Making the Frosting:
In a bowl, cream together softened cream cheese and butter until smooth.
Add in the vanilla and milk and lightly beat.
Last, add in powdered sugar one cup at a time until frosting starts to form and it is creamy and smooth.
Note: If frosting is too thin, add in 1/4 c. powdered sugar to stiffen a bit. If you want to thin frosting out, try adding 1 tsp. of milk at a time until desired consistency is reached.