In the bowl of a stand mixer fitted with a dough hook attachment (or in a large mixing bowl with a whisk), combine the water, sugar, butter, egg yolks, yeast, vanilla, and salt on medium-low speed until combined.

In a medium-sized bowl, whisk together the flour and baking powder, then gradually add to the yeast mixture until the dough is combined and pulling away from the sides of the bowl but still quite soft and sticky. Add a bit more flour, one tablespoon at a time, if necessary. (If mixing by hand, use a wooden spoon until the dough becomes too thick, then switch to your hands. Knead for 10 minutes.)

Continue to knead on medium-low speed for 5 minutes, then transfer the dough to a large, oiled bowl, cover with plastic wrap, and let rest in a warm place until doubled in size, 60-90 minutes.

Line 2 baking sheets with parchment paper, dust with a little flour, and set aside.

Transfer the dough to a lightly floured surface and using a rolling pin gently roll out to a 10 x 12-inch (25 ½ x 30 ½ cm) rectangle. Using a 3-inch (7½ cm) round biscuit cutter or cookie cutter, cut as many circles as you can.

Place the donuts on the prepared baking sheets and cover with another baking sheet to keep them from drying out. (You can also use plastic wrap or a kitchen towel but this dough is quite soft and may get stuck to whatever is touching it.)

Let rise in a warm place for 45-60 minutes, until doubled in size.

Line a baking sheet with a wire rack and place near the stove. Fill a large, heavy-bottomed saucepan with 2 inches (5 cm) of oil, clip in a candy thermometer, and turn to medium heat until the temperature reaches 350°F (180°C).

Fry the donuts, 3 -4 at a time, for about 1-2 minutes on each side, until golden brown. Let drain on the wire rack until completely cool.

Fill the donuts: Fill a pastry bag fitted with a small round tip with the pastry cream.

Use a wooden skewer or chopstick to poke a hole into the side of a donut and create a small pocket.

Squeeze a few tablespoons of pastry cream into the donut and repeat with the remaining donuts.

Place the chocolate glaze in a shallow bowl and dip the top half of the filled donuts in the glaze. Let set for about 15 minutes before serving.

These donuts are best just after they are made but you can store leftovers in an airtight container in the refrigerator for up to 1 day.