Take a mixing bowl, and mix the flour, sugar, and yeast in it, then stir until combined.
Next, evenly distribute the butter cubes over the flour mixture. Next, incorporate the eggs, milk at room temperature, vanilla essence, and a pinch of salt.
Attach the dough hook attachment to the mixer and knead the dough at a low speed for a duration of 10 minutes. If using a KitchenAid mixer, I recommend setting the speed to 2.
Upon completion of the 10-minute kneading process, the dough should exhibit a smooth, stretchy, and slightly sticky texture (please refer to recipe notes 1 & 2 for guidance on using cups as measurements).
Transfer the dough to a clean workspace without using any flour, avoiding extra flour if possible. Shape the dough into a ball by kneading it 2 to 3 times.
Place the ball of dough into a bowl that has been lightly coated with oil and wiped to be clean. Cover the bowl with plastic wrap and allow the dough to ferment for 2-3 hours, or until it has increased three times its original size.
When the dough has risen, transfer it to a clean workspace, gently remove the air, and knead it a few times until it becomes a smooth ball again.
Use your hands to flatten the dough into a ball, then use a rolling pin to shape it into a rectangle that's ½ inch (1.5 cm) thick.
To make round doughnuts, use either a cookie cutter or a glass with a diameter of 3.5 inches to cut out the dough. Any scraps can be shaped randomly for frying or re-kneaded, then re-rolled and cut out again to make more doughnuts.
Once the doughnuts have been cut, place them on a baking parchment-lined surface and cover with plastic wrap, making sure it's loose enough for them to rise. Let them prove for a time period of 1 to 1.5 hours or until they've tripled in size.
As the doughnuts are proving, prepare the pastry cream according to the instructions provided and let it cool.