Preheat oven to 350°F (175°C). Grease baking sheets or line with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, cream together the butter and sugar until light and fluffy. Add pumpkin, egg, and vanilla, and beat until well blended. Gradually add the dry ingredients to the pumpkin mixture, and beat until just combined.
Drop dough by tablespoonfuls onto prepared baking sheets. Flatten slightly with a spoon.
Bake for 10-12 minutes, or until edges are golden. Remove from oven and let cool on baking sheets for 5 minutes, then transfer to wire racks to cool completely.
For the icing: In a small bowl, mix together powdered sugar, milk, melted butter, and vanilla until smooth. Drizzle over cooled cookies.