Preheat oven to 350°F (175°C) and line baking sheets with parchment paper. In a medium bowl, whisk together the flour, baking soda, cinnamon, nutmeg, ginger, and salt; set this dry mixture aside.

In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy, then beat in the egg and vanilla extract until well combined.

Mix in the pumpkin puree until just incorporated, then gradually add the dry ingredient mixture to the wet ingredients until just combined.

Drop rounded tablespoons of dough onto the prepared baking sheets, spacing them apart. Bake for 10-12 minutes, or until edges are lightly golden and centers are set.

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

In a medium bowl, beat softened cream cheese and butter until smooth. Gradually add powdered sugar, mixing until smooth, then stir in vanilla extract.

Add milk or cream one tablespoon at a time until the frosting reaches your desired consistency. Once cookies are completely cooled, spread or pipe frosting onto each.