• Add the warm milk and stir in the yeast and sugar to the bowl of a stand mixer fitted with a dough hook. If using active dry yeast, let this sit for 5-10 minutes until foamy.
• Add the eggs, flour, vanilla and salt to the milk mixture. Turn the mixer on low speed or use your hands to combine until it forms a sticky, thick dough. Mix this dough for around 2 minutes to begin developing the gluten.
• Add the softened butter. Turn the mixer on medium speed until the sticky dough strengthens and comes together. In the mixer, it should pull away cleanly from the sides of the bowl. Mix the dough for at least 10 minutes for the best gluten development. The dough should feel smooth, soft, and strong.
• Shape the dough into a ball and place it into a large bowl. Cover the bowl tightly and let it proof in a warm spot until doubled in size. You can also proof the dough in the refrigerator overnight for extra flavor.

Shaping

• Punch down the dough and pull it from the bowl onto a lightly floured surface. If it is refrigerated, it will be stiffer.
• Form it into a mound and lightly flour the top. Then roll it into a ½ inch/ 1.3cm thick rectangle.
• Use a cookie cutter or jar lid to cut out round donuts about 3.5 inches/9 cm in diameter. Place each cut donut onto some parchment paper.
• Any dough scraps can be kneaded back into a ball and rolled out again. Leave the dough scraps to rest for a few minutes before rolling out again. This allows the gluten to relax again so the dough doesn't spring back when rolled.
• Let the donuts rest for 30-60 minutes until they are puffy. If drying out, lightly brush the tops with a little water. Use kitchen scissors to cut around each donut on the parchment paper so each donut is sitting on an individual piece of parchment.

Frying

• When nearly ready to fry, pour the frying oil into a deep fryer or a deep pot and heat it to around 340°F / 171°C - 347°F / 175°C.
• Carefully drop 1-2 donuts into the oil and remove the parchment paper with tongs. Fry the donuts for 2 minutes on one side, flip them, and fry for 2 minutes on the other side. Ensure the oil in the pot stays hot, but don't let it surpass 350°F / 177°C, or the donuts will burn before they can cook in the middle.
• Remove the fried donuts with a slotted spoon and let them drain for 20-30 seconds before placing them on paper towels to abosrb any excess oil.
• When the doughnuts have cooled, toss them in superfine sugar and fill them with your favorite fillings. Make a small hole in the side of the doughnuts and use a piping bag fitted with a small round tip to stuff them with fillings of your choice.