Whisk the dry ingredients: Put the flour, psyllium and yeast into the bowl of a stand mixer. Whisk them together on low speed to aerate the flour.
Heat the milk: Set a small pot on low heat. Pour in the milk and heat until just warm (75 degrees F if you are using a thermometer). Turn off the heat.
Make the doughnuts: With the stand mixer running at low speed, add the sugar, salt and warm milk to the flour mixture.
With the mixer still running at low speed, add 1 egg at a time until all the eggs are fully incorporated into the batter. Dribble in the vanilla extract. Scrape the batter down the sides of the bowl fully. Run the mixer until the batter is airy and light, about 5 minutes.
Add the butter: Scrape down the sides of the bowl again. With the mixer running at low speed, add the butter, one cube at a time, until fully incorporated into the batter.
At this point, the dough should gather around the paddle of the stand mixer and gently slump off when you turn the mixer off. If it feels too wet, add a bit more flour. Turn the dough onto a lightly-floured surface. Wet your hands lightly.
Gently pat the dough, which will be sticky. Don't worry. That's why you wet your hands. Gently shape the dough, stretching and pulling as you can, into a large ball. Make a seam with the edges of the dough together. Put the dough, seam-side down, into a greased bowl. Cover the bowl with plastic wrap and let it sit in a warm place in the kitchen for 1 hour.
Roll out the dough: With wet hands, gather the dough and plop it onto a lightly-floured piece of parchment paper. Flour a rolling pin and roll out the dough just shy of the edges of the parchment paper. (Remember to go slowly. This will not feel like gluten dough.) The dough should be about 1-inch thick. Put the parchment paper onto a baking sheet and place it in the freezer for 10 minutes.
Cut out the doughnuts: Using a 3-inch doughnut cutter (or biscuit cutter or an empty tin can, if that's all you have), cut out 8 rounds from the dough. Cut holes in the rounds with a much smaller cutter. (You could also use your finger.) Set them aside on a piece of parchment paper.
You can roll out the remaining dough and cut out more doughnuts, if you want. They won't be quite as light as the first batch, but they are still doughnuts!
Let the doughnuts rest while you prepare the oil.
Heat the oil: Set a Dutch oven over medium-high heat. Pour in enough oil to fill the pot 1/3 of the way. Heat the oil to 350 degrees F. Fry the doughnuts: Using a slotted spoon, gently lower a doughnut into the hot oil. Add three more doughnuts.
Fry the doughnuts for 30 seconds on one side, then 45 seconds on the other side, then another 30 to 45 seconds on the first side again, until the doughnuts are a deep golden brown.
Remove the doughnuts to a baking rack set over a baking sheet. Fry the remaining doughnuts the same way.
Make the glaze: Whisk together the milk and vanilla extract in a medium bowl. Continue whisking as you add the confectioners' sugar. Whisk until all the lumps of sugar have been blended into a smooth glaze.
Dunk the warm doughnuts into the sugar glaze and return them to the cooling rack.
Eat immediately. Doughnuts are really only good the day they are made and they're best within an hour of being made.
Gluten-Free Pastry Flour Mix:
Pour all the flours and milk powder into a large container. (Restaurant supply stores sell large plastic containers that fit this purpose well. You could also use a large glass jar.) Shake and shake and shake harder until all the flours have become one color.