Preheat your oven to 350°F (175°C). Line cookie sheets with parchment paper.
In a large bowl, cream together the softened butter and sugar until light and fluffy. Beat in the pumpkin puree, egg, and vanilla extract until well combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Drop the dough by tablespoonfuls onto the prepared cookie sheets, spacing them about 2 inches apart. Bake in preheated oven for 12-15 minutes until the edges are lightly browned.
Remove from oven and allow to cool on the baking sheets for a few minutes before transferring to a cooling rack to cool completely.
In a small saucepan, combine the powdered sugar, milk, melted butter, and vanilla extract. Heat over low heat, whisking continuously until the mixture is smooth and well combined.
Once the cookies have cooled, spoon the icing over each cookie, spreading it to cover the top. Allow the icing to set for about 30 minutes before serving.