Place yeast, warm water, and and 1/2 tsp sugar in a bowl to proof. Let it sit for 5-10 minutes, until frothy and bubbly. (I do this in a stand mixer but you can mix by hand as well).
In a separate bowl, whisk together 2 cups of flour and salt until combined.
To the yeast mix, add milk, the rest of the sugar, egg, butter OR oil(I've tried it with both and they tasted the same), and vanilla extract. Add flour mix and stir to combine. The dough will be shaggy. Gradually, add the 3/4 cup of flour and mix until the dough comes together.
Sprinkle your work surface lightly with flour and turn the dough out. Knead for a couple of minutes, until the dough is smooth and no longer sticky. Transfer to an oiled bowl(or a bowl sprayed with cooking spray), cover with a towel, and let rise for an hour.
Sprinkle your work surface again with flour(so the doughnuts don't stick to the counter), pop out the dough, and sprinkle the top with flour as well. Roll it out to about 1/2 inch thick. Using a round cookie cutter or the lid of a jar, cut out doughnut shapes, pressing firmly into the dough to cut all the way around the edges. Using a smaller jar or cutter, make a hole in the center, and keep those middles holes!! Place all doughnuts and holes on a parchment lined baking tray, cover with a towel, and let rest for at least 30 minutes.
If you want to make these in advance, stop now. Place the tray of covered donuts into the fridge. They can stay there for up to TWO DAYS. When you are ready to fry them, simply pop the donuts directly into hot oil and continue.