In a large bowl, whisk together the flour, sugar, baking powder, salt, and pumpkin pie spice.
In a separate medium mixing bowl, whisk together the pumpkin puree, buttermilk, eggs, melted butter, and vanilla extract, and pumpkin puree.
Pour the wet mixture into the dry and mix until the dough is combined. (The dough will still be quite sticky.)
Turn the dough out onto a well-floured surface and roll out to about 1/2 inch thickness. (The donuts will puff up while frying.)
Use a donut cutter to cut out donuts and donut holes.
Heat 3-inches of oil in a large skillet or pot over medium high heat. When the oil reaches 350F, carefully slide a few donuts into the oil, frying each side until lightly browned. Fry the donuts in small batches until they’re all cooked.
Cool the donuts slightly on a wire rack.
In a medium bowl, mix together the ingredients for the glaze until smooth. Dip each warm donut into the glaze and place back on the wire cooling rack for excess glaze to drip off.
This is a very soft dough. When frying the donuts, I find it best to place each donut on a heat-safe metal slotted spoon or spatula, lower it gently into the oil, and scoot it off into the oil. This is easier than picking up the donuts and placing them into the oil by hand or with tongs because they end up stretching, tearing, or become really misshapen.
Tongs are best to carefully flip the donuts in the oil for the best control.