Heat the oil in a large saucepan to 375 degrees F.

In a medium bowl, whisk the buttermilk, pumpkin, white sugar, and eggs.

Mix flour, baking soda, baking powder, salt, and pumpkin pie spice together in another large bowl.

Add the buttermilk mixture into the flour mixture, and stir or knead with your hands until combined.

Add melted butter and knead until the dough is soft.

Roll dough onto a lightly floured surface, to ¼-inch thick.

Use a donut cutter to cut the donuts.

Carefully add donuts to the pan of oil, using a slotted metal spoon.

Cook until golden brown, about 1 minute, then carefully flip over, and cook for an additional minute on the other side.

Transfer to a cooling rack with a baking pan underneath.

Stir the powdered sugar, vanilla extract, and buttermilk in a bowl until smooth.

Dip the donuts one by one into the glaze, then set back on the cooling rack.