Line a rimmed baking sheet with paper towels and set a wire rack on top. Pour 6 cups of oil into a Dutch oven and clip on a thermometer.
Heat the oil over medium heat to 350°F, which takes 10 to 15 minutes. Start this before mixing the batter.
In a large bowl, whisk together the flour, baking powder, pumpkin pie spice, and salt for about 30 seconds. Set aside.
In a medium bowl, whisk the brown sugar, pumpkin puree, buttermilk, and egg until smooth. Whisk in the melted butter and 1 teaspoon vanilla until glossy.
Pour the wet mixture into the dry ingredients and fold with a rubber spatula until no dry flour streaks remain. Do not overmix.
Use a cookie scoop to gently lower 5 or 6 scoops of batter into the 350°F oil. Fry for 3 to 4 minutes, turning often, until deep golden brown.
Lift the donut holes out with a spider strainer and transfer to the wire rack. Let the oil return to 350°F and repeat with remaining batter.
In a wide shallow bowl, whisk the powdered sugar, 1/2 teaspoon pumpkin pie spice, 1/2 teaspoon vanilla, and 3 tablespoons milk until smooth.
Roll a few warm donut holes in the glaze until lightly coated. Lift out, let excess drip off, and return to the wire rack.
Let the glaze set for about 10 minutes, then pile onto a platter and serve while still soft and slightly warm.