Pour 3/4 cup of warm milk into a small bowl. Sprinkle 2 and 1/4 teaspoons of active dry yeast over the milk. Add a pinch of sugar and stir gently.

Let the mixture sit for about 8 to 10 minutes until it foams up.

In a large mixing bowl, whisk together 3 cups of flour, 1/4 cup of sugar, and 1/2 teaspoon of salt. Make a well in the center and add the foamy yeast mixture, 2 eggs, 1 teaspoon of vanilla extract, and 4 tablespoons of softened butter.

Mix until no dry flour remains.

Turn the dough out onto a lightly floured surface and knead for 8 to 10 minutes until smooth and elastic.

Shape the dough into a ball, place it in a lightly oiled bowl, cover, and let it rise for 60 to 75 minutes until doubled in size.

Punch the dough down, roll it out to about 1/2 inch thickness, and cut out doughnuts using a 3-inch round cutter.

Cover the cut doughnuts loosely and let them rise for 30 to 45 minutes.

In a medium saucepan, heat 2 cups of whole milk until just steaming.

Whisk together 4 egg yolks, 1/2 cup of sugar, and 3 tablespoons of cornstarch, then gradually add the hot milk.

Return to the saucepan and cook over medium heat until thickened, then mix in 2 tablespoons of butter and 1 teaspoon of vanilla extract.

Whip 1 cup of heavy whipping cream to stiff peaks and gently fold it into the cooled pastry cream.

Heat oil to 175°C (350°F) and fry 2 to 3 doughnuts at a time for about 1 minute 30 seconds per side.

Use a piping bag to fill each cooled doughnut with the cream filling, then dust with powdered sugar.