In a large bowl of standing mixer, add lukewarm milk and dry active yeast. Sprinkle with 1 tsp of sugar and whisk with hand whisker and let stand for 5 minutes.
Add to the milk cooled melted butter, egg, pumpkin purée, vanilla extract, pumpkin pie spice and salt. Whisk until all ingredients are well combined.
Place the large bowl onto the standing mixer with attached dough hook and start adding slowly flour (cup by cup). Once the dough is formed, continue kneading in the mixer for another 5 minutes or until the dough is soft. The dough will be slightly sticky.
Leave the dough in the mixing bowl, cover with kitchen towel and let it prove in a warm place, for 1 and 1/2 hours or until the dough has doubled.
Once the dough has proofed, remove it from the bowl on flour dusted working surface and roll the dough with rolling pin, about 1/3 inch thick. Cut out round shapes using classic doughnut cutter or 3" circle cutter.
To make the middle hole, use the other end of large piping tip or small cookie cutter around 1 inch large. When all the donuts are cut out, collect the rest of the dough, knead and reuse. Place the shaped donuts on baking sheet covered with parchment paper. Cover with kitchen towel and let proof for another 20 minutes.