Heat oven to 350ºF.
In a mixing bowl, combine the oil, eggs, and vanilla. Beat in the sugars, then stir in the pumpkin puree.
In a second bowl or large measuring cup, stir together the remaining ingredients. Stir the dry ingredients into the wet ingredients until there are no lumps.
Spoon 1/4 cup of batter into each well of a 12-donut pan, as neatly as you can (you can also use a pastry bag to pipe it into the pan). Bake for 20 minutes, or until a toothpick inserted into a donut comes out clean. Let the donuts cool, then remove from the pan.
While the donuts cool, prepare the frosting. Use an electric mixer to beat the butter until it's light and fluffy. Add the sugars, vanilla, and milk and beat until the frosting is light and fluffy. If necessary, you can thin the frosting out with a little more milk.
Pipe the frosting onto the cooled donuts. Top with chopped pecans, if desired.