Activate the yeast. In the bowl of your stand mixer fitted with the paddle attachment (see note), mix together the warm milk, sugar, and yeast. Let sit for 10 minutes while the yeast activates and foams. Note: If your yeast hasn't foamed up after 10 minutes, you may have bad yeast and you should try again with a new packet.

Mix dry ingredients. In a medium bowl, whisk together all of the dry ingredients: oat flour, millet flour, sweet rice flour, tapioca flour, xanthan gum, spices, kosher salt, and baking powder.

Make the dough. Once the yeast has foamed, add the pumpkin, eggs, melted butter, vanilla, and about 1 cup of the dry mixture to the stand mixer and mix until completely smooth, scraping down the sides as needed. With the mixer on low, slowly mix in the remaining dry ingredients, scraping down the sides as needed. Mix on high speed for 2 minutes. The dough will be very soft and sticky.

Toss in tapioca flour and roll out the dough. Coat your work surface well with tapioca flour. Use a spatula to dump the dough onto your work surface. Sprinkle more tapioca flour on top of the dough and ensure your hands are coated in tapioca flour before touching the dough. This dough is much softer and sticker than gluten dough. Gently form the dough into a ball with the surface of the dough entirely dusted with tapioca flour so it is no longer sticky. Roll the dough into ½ - ¾ inch thickness.

Cut out your donuts. Use a sharp round cutter to cut out your donuts and/or donut holes. I use 3 to 3.5-inch cutters with 1.5-1.75-inch holes. Use a bench scraper to transfer each donut to a parchment-lined baking sheet, at least 2 inches apart. For donut holes, roll the dough into a ball with the palms of your hands. With tapioca-floured hands, knead the scraps into a ball and repeat until you've made all the donuts.

Let the dough rise. Cover with plastic wrap and let rise in a warm place (such as the proofing setting on your oven) for 1-1.5 hours. Note: if making ahead, you can skip this rise and place the covered donuts directly in the fridge overnight. The next day, pull the baking sheet out of the fridge and let it sit on the counter for an hour before frying and continue as written.

Prepare the glaze, sugar, or icing. Whisk together the glaze or icing ingredients or set a bowl of sugar for tossing fried donuts in. 

Fry the donuts. Heat at least 2 inches of oil to 360°F. This is a bit cooler than normal frying temperature because the gluten-free dough needs to cook just a little longer and you don't want the outside to get too dark in the process. Use a candy thermometer to keep the oil at a consistent temperature. Place a wire rack over a baking sheet nearby. Carefully drop your donut or donut hole into the oil. Fry 3-4 donuts at a time, 2 minutes on each side, until golden brown. Transfer to the prepared rack to cool slightly for 3-5 minutes. Repeat.

Glaze the donuts. When you finish frying the next batch, dunk the previous batch of slightly cooled donuts into the glaze or bowl of sugar on both sides. Place back on the wire rack. The glaze will set after about 20 minutes. Optionally dust the sugar donuts in cinnamon or pumpkin spice. If doing thicker icing with sprinkles, let the donut cool 10 minutes before dunking in the icing.

Enjoy! For best results, serve donuts the same day. If you have leftovers, store them in an airtight container for up to a day and warm them in the microwave for 10-15 seconds, or better yet, toss them in the air fryer at 400°F for 2 minutes, before serving.