Preheat the oven to 350 degrees. Prepare two large baking sheets with parchment paper. In a large bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
In another mixing bowl, beat the butter and sugar until combined and fluffy, about 6 minutes. To the bowl, add the pumpkin puree and beat until evenly incorporated, scraping down the sides as needed.
Then, mix in the egg and vanilla extract. To the wet ingredients, add the flour mixture and beat just until combined. Using a small cookie scoop or two spoons, drop balls of cookie dough onto the prepared cookie sheet about 2 inches apart.
Bake for 15 to 18 minutes or until edges are hardening and middle inside is no longer wet but still tender. Cool for 5 to 7 minutes on the baking sheet before transferring to wire racks.
To make glaze; Use a hand or stand mixer on medium speed with cream cheese to mix and get smooth. Add other ingredients, add more milk to thin out (you want it a bit thick but able to squeeze out of a pastry bag).