Preheat the oven to 350°F (177°C). Line baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, beat the butter, oil, granulated sugar, and powdered sugar together until smooth.
Add the pumpkin, egg, and vanilla extract, and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until fully incorporated.
Drop rounded tablespoons of dough onto the prepared baking sheets.
Bake for 10-12 minutes, or until the edges are set. Cool on the sheet for 2 minutes, then transfer to a wire rack.
For the glaze, whisk powdered sugar and milk until smooth. Drizzle over cooled cookies.