With the mixer still running, add the eggs, vanilla, and pumpkin. Mix until the ingredients are uniformly combined. Add 1 cup of flour at a time, mixing well after each addition and scraping down the sides of the bowl. If you cannot fit all the flour in your standing mixer bowl, stir in any remaining flour by hand.
In an electric mixer fit with the paddle attachment, cream the butter on medium-high speed for 30 seconds. Add sugar, baking powder, baking soda, cinnamon, nutmeg, and salt. Continue mixing until all the ingredients have been incorporated, scraping down the sides as necessary.
Adjust an oven rack to the top middle position and preheat oven to 350°F.