Add oil to a Dutch oven or a fryer and heat it over medium heat until oil reaches 350° degrees.
Line a large cookie sheet with paper towels to catch oil and place a cooling rack on top of paper towels.
In a medium mixing bowl, whisk buttermilk, pumpkin puree, vanilla extract, and egg until smooth. Whisk in melted butter.
Add flour, baking powder, pumpkin pie spice, dark brown sugar, and salt. Whisk until the ingredients are completely incorporated.
When the oil reached 350°, use a cookie scoop (#50) to scoop out dough and drop it into heated oil in batches of 5-6, depending on how big your pot is. TIP: I dip the cookie scoop into some oil and shake it off before doing each batch, this prevents dough from sticking and created smooth dough drops. I also rinse out the scoop between batches, to wash off any stuck on dough. Again, this is to help create smooth doughnut holes.
Fry doughnut holes for about 4-5 minutes (until they turn deep golden), turning them over every couple of minutes if they haven't turned themselves. (Remember to use metal tongs or metal slotted spoon to take doughnuts out and turn them.)
Take doughnuts out onto the prepared cooling rack. Repeat frying the dough in batches with remaining batter.
Mix powdered sugar and pumpkin pie spice. Once the doughnut holes are cooled enough to handle, dust them with the powdered sugar mixture.