Add the warm milk to the bowl of a stand mixer fitted with a dough hook, then stir in the yeast and sugar. If using active dried yeast, let it sit for 5 to 10 minutes until foamy.

Add the room-temperature butter. Turn the mixer to medium speed until the dough strengthens and comes together. It should pull away cleanly from the sides of the bowl. Mix 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 in a lightly greased large bowl. Cover and proof in a warm spot until doubled in size. You can also proof the dough in the fridge overnight for a deeper flavour.

In a medium saucepan, heat the milk until it's just at a simmer, but don't let it boil.

Return the combined mixture to the saucepan over medium heat, whisking constantly until it bubbles. Once it starts bubbling, it will begin to thicken. Keep it at a bubble, and don't stop whisking, for 1 minute.

Take it off the heat and whisk through the butter until combined. Pour the hot pastry cream into a clean bowl. If you'd like an extra silky smooth texture, push it through a fine mesh sieve first.

Lay a piece of plastic wrap directly onto the surface of the custard to stop a skin forming. Let it come to room temperature, then chill in the fridge until completely cold. The pastry cream can be made up to 2 days in advance.

Gently deflate the dough and turn it out onto a lightly floured surface. If it's been refrigerated it will be stiff. Form it into a mound, lightly flour the top, then roll into a 1.3cm (½ inch) thick rectangle.

Use a cookie cutter or jar lid to cut out 9cm (3½ inch) rounds. Place each doughnut on baking paper. Knead any scraps back into a ball, rest for a few minutes (so the gluten relaxes and the dough doesn't spring back), then roll out and cut again.

Let the doughnuts rest for 30 to 60 minutes, until they're puffy. If they look like they're drying out, lightly brush the tops with water. Use kitchen scissors to cut around each doughnut on the baking paper so each one sits on its own piece.

When nearly ready to fry, pour the oil into a deep fryer or deep pot and heat to 170°C / 340°F. Fill a shallow bowl with the coating sugar.

Carefully drop 2 to 3 doughnuts into the oil and remove the baking paper with tongs. Fry for 2 minutes until deep golden brown, flip, and fry for another 2 minutes.

Keep the oil hot but don't let it go above 175°C / 350°F, or the doughnuts will brown before the middles cook through.

Give the cold custard a whisk to loosen it up, then spoon it into a piping bag with a round tip. Insert the tip of a small knife into the side of each cooled doughnut to make a hole for the filling. Pipe the custard through the hole into each doughnut.