Prepare three baking sheets, one lined with parchment paper, one with paper towels, and a third with a wire rack. Set aside.
In a large bowl, whisk together flour, sugar, baking powder, pumpkin pie spice, and salt.
In a second bowl, whisk together pumpkin, buttermilk, eggs, melted butter, and vanilla extract.
Pour the wet ingredients into the dry ingredients; Stir by hand until well combined.
Pour the dough out onto a floured surface. Roll out the dough to ½-inch thickness, adding more flour to the work surface as needed. (The donuts will puff up when cooked in oil.)
Once rolled out, use a donut cutter to cut out shapes. If you do not have a donut cutter, use a round tool like a kitchen glass or a jar lid that is about 3-inches across. You can use a bottle cap or small shot glass for the smaller hole.
Set cut-out donuts on the parchment-lined baking sheet. Set aside while you prepare the glaze.