In the bowl of a stand mixer, whisk together the milk, eggs, and egg yolk until combined. Add the bread flour, sugar, salt, and yeast. Mix with the dough hook on low speed for about 5 to 7 minutes until a rough but cohesive dough forms.

Cover the bowl and let the dough rest for 5 minutes. This short pause helps the gluten strengthen naturally and keeps the dough from overheating.

Resume mixing and gradually add the softened butter, a few pieces at a time, allowing each addition to fully incorporate before adding more. Continue kneading until the dough becomes smooth, elastic, and stretches into a thin "windowpane" without tearing. Shape into a ball, return it to the bowl, cover, and refrigerate for 12 to 20 hours.

Line baking trays with parchment squares. Divide the chilled dough into 13 equal portions and roll each into a smooth ball. Place each ball on its own parchment square and gently flatten into discs about 3½ inches wide. Cover loosely and let rise in a warm place until doubled in size and very puffy. This usually takes between 1½ and 3 hours.

While the dough is rising, prepare the filling. In a saucepan, whisk together the egg yolks, sugar, cornstarch, and salt. Slowly whisk in the half-and-half until smooth. Cook over medium heat, whisking continuously, until the mixture thickens and boils. Continue cooking for another minute. Remove from the heat and stir in the butter and vanilla.

Transfer to a shallow dish, cover the surface directly with plastic wrap, and chill until completely cold. Before using, stir the chilled pastry cream until smooth. Whip the heavy cream until stiff peaks form, then gently fold it into the pastry cream in two additions to create a light, airy crème légère. Keep refrigerated until ready to fill the doughnuts.

Heat about 3 inches of oil in a heavy pot to 350°F (175°C). Carefully lower two doughnuts into the oil with the parchment attached. After a few seconds, remove the parchment using tongs.

Fry for 1 to 2 minutes per side until deep golden brown and cooked through. Transfer to a cooling rack lined with paper towels.

While still warm, roll each doughnut in granulated sugar until evenly coated.

Repeat with the remaining dough, allowing the oil to return to 350°F between batches.

Transfer the chilled crème légère into a piping bag fitted with a large round opening. Use a small paring knife to make an opening in the side of each doughnut, gently twisting to create room for the filling. Insert the piping bag and squeeze generously until the doughnut feels full. Pipe a small swirl of cream at the opening if desired.

Serve immediately for the freshest taste.