Check yeast instructions:

Inspect the yeast package: certain yeasts can be incorporated directly with dry ingredients as described; others require activation in a small amount of warm liquid. If activation is needed, use a portion of the warm milk (do not add extra liquid).

Mix dry ingredients:

In a mixing bowl, stir together the flour, sugar, and yeast until well blended.

Add butter and eggs:

Scatter the softened butter cubes over the flour mixture. Add the eggs, room-temperature milk, vanilla paste, and salt.

Knead dough 10 minutes:

Fit the dough hook and knead on low speed (KitchenAid speed 2) for 10 minutes. The dough should be smooth, elastic, and slightly tacky.

Form dough ball:

Move the dough to a clean work surface (avoid additional flour). Form into a ball by kneading 2-3 times.

First rise 2-3 hours:

Put the dough ball into a lightly oiled bowl, cover with plastic wrap, and let rise for 2-3 hours until tripled in volume.

Deflate and knead:

Turn the risen dough onto a clean surface, gently press out gas, and knead a few times until smooth again.

Roll dough rectangle:

Flatten the dough with your hands, then roll into a 1/2-inch (1.5 cm) thick rectangle.

Cut out doughnuts:

Cut out round doughnuts using a 3.5-inch cookie cutter or glass. Re-knead and re-roll scraps to produce additional doughnuts.

Proof doughnuts 1-1.5 hours:

Arrange cut doughnuts on a parchment-lined surface, loosely cover with plastic wrap, and proof for 1-1.5 hours until tripled in size.

Prepare pastry cream:

While doughnuts proof, prepare the pastry cream (see below) and allow it to cool.

Heat oil to 337°F:

Heat oil in a large deep pan to 337°F (170°C).

Cut parchment squares:

Cut parchment into squares, one per doughnut.

Lower doughnuts into oil:

Carefully lower doughnuts into hot oil with tongs, removing the parchment promptly. Fry in small batches to avoid overcrowding.

Fry 2 minutes per side:

Fry 2 minutes per side, turning once. Drain on paper towels, then roll in sugar. Cool before filling.

Create hole for filling:

To fill, create a small hole in the side of each doughnut with a knife, twisting to form a pocket.

Pipe in pastry cream:

Load a piping bag with pastry cream and pipe into the doughnuts.

Serve or store:

Serve immediately, or store in a plastic container for up to 1-2 days.

Whisk yolks and sugar:

For the pastry cream: In a large bowl, whisk egg yolks, sugar, vanilla paste, and lemon zest until uniform.

Add cornstarch:

Whisk in cornstarch until smooth.

Heat milk until hot:

Heat milk in a saucepan until hot, then remove from heat and let cool for 1 minute.

Temper egg mixture:

Gradually pour the hot milk into the egg mixture, whisking constantly to prevent curdling.

Cook cream until thick:

Pour the mixture back into the saucepan and cook over low heat, stirring continuously, until thick and creamy (10-15 minutes). Do not raise the heat or the eggs may curdle.

Cool cream with plastic:

Once thickened, transfer to a clean bowl and press plastic wrap directly onto the surface to prevent a skin. Let cool completely.