Mix milk, yeast, sugar:

Pour the lukewarm milk into a stand mixer bowl with a dough hook, then mix in the yeast and sugar. If using active dried yeast, allow it to rest for 5 to 10 minutes until frothy.

Add eggs, flour, vanilla:

Add the eggs, flour, vanilla, and salt to the milk mixture. Run the mixer on low speed, or combine by hand, until a stiff dough forms. The dough will be quite firm and tight initially; this is expected, as it will soften after the butter is added. Mix for about 2 minutes to begin gluten development.

Incorporate softened butter:

Incorporate the softened butter. Increase the mixer to medium speed and beat until the dough becomes elastic and pulls away from the bowl sides. Continue mixing for at least 10 minutes to achieve optimal gluten structure. The dough should feel smooth, pliable, and resilient.

Let dough rise:

Form the dough into a ball and transfer it to a lightly greased large bowl. Cover and let it rise in a warm environment until doubled in size. For a more developed flavor, refrigerate the dough overnight.

Warm milk for custard:

In a medium saucepan, warm the milk until it just begins to simmer; avoid boiling.

Whisk egg yolk mixture:

As the milk heats, whisk together the egg yolks, sugar, salt, vanilla, and cornstarch in a separate heatproof bowl until the mixture is smooth.

Temper egg mixture:

Set the bowl on a damp kitchen towel to prevent slipping. Slowly pour the hot milk into the egg mixture while whisking continuously.

Cook custard until thick:

Return the combined liquid to the saucepan and place over medium heat, whisking constantly until it begins to bubble. Once bubbling, the custard will thicken. Continue whisking steadily for 1 minute while it bubbles.

Add butter, strain custard:

Remove from heat and whisk in the butter until fully incorporated. Transfer the hot pastry cream to a clean bowl. For an exceptionally smooth texture, press it through a fine-mesh sieve.

Chill pastry cream:

Press a piece of plastic wrap directly onto the custard's surface to prevent a skin from forming. Allow it to cool to room temperature, then refrigerate until fully chilled. The pastry cream can be prepared up to 2 days ahead.

Roll dough into rectangle:

Gently punch down the dough and turn it out onto a lightly floured surface. If the dough was refrigerated, it will be firm. Shape it into a mound, dust the top with flour, and roll it into a rectangle 1/2 inch (1.3 cm) thick.

Cut doughnut rounds:

Using a cookie cutter or jar lid, cut out rounds 3 1/2 inches (9 cm) in diameter. Place each doughnut on a piece of baking paper. Gather the scraps, knead them into a ball, let rest for a few minutes (to relax the gluten and prevent spring-back), then roll and cut again.

Rest doughnuts until puffy:

Allow the doughnuts to rest for 30 to 60 minutes until they become puffy. If they appear dry, lightly brush the tops with water. Using kitchen scissors, cut around each doughnut on the baking paper so each sits on its own square.

Heat oil for frying:

Shortly before frying, pour oil into a deep fryer or deep pot and heat to 340°F (170°C). Fill a shallow bowl with the coating sugar.

Fry doughnuts 2 minutes per side:

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

Monitor oil temperature:

Maintain the oil temperature, ensuring it does not exceed 350°F (175°C), or the doughnuts will brown on the outside before the centers are cooked.

Coat doughnuts in sugar:

Remove the fried doughnuts with a slotted spoon and briefly place them on a wire rack. While still hot, toss them in a bowl of granulated or caster sugar to coat, then return to the rack and continue frying the remaining doughnuts.

Fill doughnuts with custard:

Whisk the chilled custard to loosen it, then spoon it into a piping bag fitted with a round tip. Insert a small knife tip into the side of each cooled doughnut to create a hole. Pipe the custard through the hole to fill each doughnut.