Measure out your vegan butter and leave it at room temperature to get soft.
In the bowl of a stand mixer, whisk together the flour, sugar, salt, and yeast.
In a saucepan, add the milk (or water) and heat to about 110°F or to warm, but not scalding. A temperature too hot could kill your yeast. Add in aquafaba.
Add it to your flour mixture and with a dough hook mix on medium speed until a a dough forms. If too dry, add some more milk or if too wet, add some flour.
In portions, slowly add sections of vegan butter, so it combines in the dough. Don't add too much at a time because the dough won't grab onto the dough hook and it won't mix properly. Only a tsp at a time is enough. Wait for it to mix into the dough, then add a little bit more.
Once the dough is done, cover the bowl let proof in a warm place for 1-2 hours. Punch down and then put in the fridge overnight.
Put on a lightly floured work surface and roll out until about 3/4" thick. Cut with a 3" round cutter or biscuit cutter.
Cut out little squares out of parchment paper of 4" by 4" and place each dough circle on them. This will make it easier to transfer to the frying oil.
Let it proof in a warm place for 30 minutes-2 hours until swollen in size.
In a heavy bottomed pot, heat 2 inches of oil to 360-375°F. Using a slotted spoon or spider, place the dough with the parchment into the oil. Remove the parchment square with tongs.
Fry donuts 2-3 at a time. Not too much because it can overcrowd the pot and make the temperature drop drastically.
Fry on each side a few minutes until golden brown with a white ring in the middle.
Let cool and then use as you want. Fill it with vegan pastry cream, jam, or chocolate. Coat in powdered sugar or a lemon glaze or chocolate glaze. Let cool before filling.
I did my donuts Boston cream style and filled with pastry cream and chocolate glaze on top.