In a large mixing bowl, whisk together the flour, sugar, baking powder, salt, and pumpkin pie spice.
In a separate mixing bowl, whisk together the pumpkin puree, buttermilk, eggs, melted butter, and vanilla extract.
Combine the wet and dry ingredients in one mixing bowl, and mix until the dough is well combined. (The dough will still be very sticky.)
Turn the dough out onto a floured surface and roll it out to about ½ inch thickness. (The finished donuts will be thicker, as they will puff up while frying.) Use a donut cutter to cut out donuts and donut holes.
Heat a few inches of oil in a large skillet over medium high heat. When the oil is hot (about 350°F), gently slide a few donuts into the oil, frying each side until lightly browned. Fry the donuts in small batches until they’ve all been cooked.
Cool the donuts slightly on a wire rack.
While the donuts are cooling a bit, mix all the ingredients for the glaze and whisk until smooth. Dip each warm donut into the glaze and place back on the wire cooling rack, allowing the excess glaze to drip off.
The dough for these pumpkin donuts is very soft, so I usually place each donut on a metal slotted spoon or spatula, lower it gently into the oil, and gently slide it off into the oil. This is easier than picking up the donuts with your hand and placing them into the oil because they will end up stretching or tearing in the process.
Use tongs to carefully flip the donuts while in the oil for the best control.