In a large mixing bowl, cream together the butter, sugar and brown sugar. Add the eggs one at a time and the vanilla. Sift the flour, baking powder and soda into a medium bowl.
Add the pumpkin, cinnamon and nutmeg to the mixing bowl. Alternate adding the buttermilk and dry ingredients until everything is combined. Use a rubber spatula to scrape the sides and bottom of the mixing bowl occasionally.
Turn the dough out onto a floured work surface. Roll out to about 1/2-inch thick and use a doughnut cutter to cut out the doughnuts. Place the doughnuts and doughnut holes on sheet pans lined with parchment paper.
Line a sheet pan with several layers of paper towels and set aside.
In a large pot, heat about 3-inches of canola oil to 375 degrees F. Use a thermometer to test the temperature. Gently drop about 4 doughnuts into the hot oil, do not over crowd the pot. Cook for about 1-2 minutes on each side. Flipping the doughnut once.
Remove the doughnuts from the oil and let drain on the paper towels.
While the doughnuts are still warm, coat with the cinnamon sugar mixture.