Place the butter and milk into a measuring 2 cup and microwave for 30 seconds until warmed, throw in the butter, stir for about 30 seconds, until butter melts, add the dry yeast, sugar, swirl gently until they are mixed in the milk/butter mixture, Let sit until foamy, about 10 minutes.
In a stand mixer add the flour, pumpkin pie spice, and salt into the bowl and with a hand whisk mix for couple of minutes.
Add the yeasty liquid, the pumpkin puree, and the egg into the dry ingredients and mix with the bread hook attachment. Use a rubber spatula to scrape down the edges as needed.
Once dough has formed in the center of the bowl, knead for 10 minutes. Cover with plastic wrap and let rest in a warm place until doubled in size, about 1 hour.
Once raised, punch down the dough and roll out to about 1 inch on a well floured surface, Use a donut cutter (3”inch) round cookie cutters, to cut out donuts. Cover loosely with a cloth and let rise (30 minutes) while heating the oil for frying.
Heat the oil in a medium saucepan over medium heat for about 15 to 20 minutes (350 f.).
Place the donuts, into the hot oil. Let it fry for 1-2 minutes on each side until golden brown, Let it sit on the paper towels for a minute to absorb some of the excess oil and cool before glazing and/or frosting.
Whisk together the confectioners’ sugar, milk, and vanilla until smooth. Add more milk (by the teaspoon) if you want the glaze to be thinner. Dunk the donut in the glaze (both sides), and place on a wire rack (with paper towels underneat