• Add the warm buttermilk and stir in the yeast and sugar to the stand mixer bowl fitted with a dough hook. If using active dry yeast, let this sit for 5-10 minutes until foamy.
• Add the remaining dough ingredients and fit the mixer with a dough hook. Turn the mixer on low and combine until it forms a thick but slightly sticky dough. Once combined, keep mixing on medium speed for 8-10 minutes until the dough is smooth, glossy, and elastic.
• Kneading by hand - Use a wooden spoon or fork and mix all the dough ingredients in a medium bowl until they form a shaggy dough ball. Once it forms a dough ball, pull it onto a floured surface and knead by hand for around 10-15 minutes until it becomes a strong and smooth dough. If you need a break, take it. The dough responds well to resting time.
• Place the dough ball in a lightly greased large bowl. Cover the bowl with plastic wrap or a lid and refrigerate for 4 to 24 hours. The longer it’s left, the more flavor they will have. If you want to skip the cold-proof, let the dough double at room temperature (around 1- 1 1/2 hours). Then, continue with the shaping.

Shaping

• Punch the dough down and pull it from the bowl onto a floured work surface or parchment paper.
• Using a rolling pin, roll the dough into a ½ inch / 1.3cm thick square. Use a doughnut cutter or or a jar lid or glass to cut out the shapes.
• Place the shaped doughnuts on parchment paper. Use kitchen scissors to cut around each so that each doughnut is sitting on an individual piece of parchment. The doughnuts can now be transported into the oil on parchment paper, which makes transferring them easier. If you don't have parchment paper, you can transfer the dougnuts without it.
• Any remaining dough can be rolled out and cut again.

  Bombolini-style doughnuts - You can also simply cut them as circles and use a large jar lid or cookie cutter to cut circles of 3.5 inches/9 cm in diameter without a hole in the middle. These will be cooked slightly longer and at a lower temperature than those with a dougnut hole removed from the middle.
• Leave the doughnuts to rise slightly for around 30-45 minutes until they have puffed a bit. This timing is dependent on room temperature – the warmer the room, the faster they’ll rise.
• When they have risen, pour the frying oil into a deep fryer or pot and heat it to 356°F/180°C. The oil in the pan should be at least 3 inches deep.

  Note- if your doughnuts were cut bomboloni-style, your oil should be less hot, 340°F / 171°C as they will take longer to cook in the middle so you don't want them going brown too fast on the outside.
• When the oil is hot, fry the doughnuts in it for 2-3 minutes, around 60-90 seconds per side. Use tongs to remove the pieces of parchment paper from the oil. It will be much less for the smaller donut holes.

  Bomboloni-style donuts will need around 2 minutes on each side.

  Don't add too many at a time because they will lower the oil temperature. If it drops too much your donuts won't fry well and will absorb too much oil. Don't let the temperature surpass 375°F / 190C.
• After frying, use a slotted spoon to remove the doughnuts from the oil, then place them on paper towels to drain.

Topping

• If using a cinnamon topping, combine the granulated sugar and cinnamon in a small bowl.
• Add the doughnuts into the sugar bowl and coat them on all sides.
• If you are glazing them in a bowl, combine powdered sugar and vanilla enough to create a thick but pourable glaze.

  When the doughnuts come from the oil, place them on paper towels to soak up excess oil. Let them cool on a cooling rack for 5 minutes, then dip the still-warm donuts into the glaze. Let them dry on the rack.