In a large bowl, mix 4 1/2 cups flour, 2 1/4 tsp yeast, 3 Tbsp sugar, and 1 tsp salt.

Add 2 large eggs, 2 large egg yolks, 1/2 cup softened butter, and 1 cup warm water.

Using a stand mixer with a dough hook, knead for 10–15 minutes until the dough is smooth and elastic.

Transfer to a greased bowl, cover, and let rise for 1–2 hours until doubled in size.

Punch down the risen dough and turn it onto a lightly floured surface.

Divide the dough in half, roll each half into a log, and cut into equal pieces.

Roll each piece into a tight ball and arrange them on a parchment-lined tray.

Cover and let proof for 30–60 minutes until slightly puffed.

Heat oil in a deep pot or fryer to 360°F (182°C).

Fry the dough balls in batches for 1–2 minutes per side until golden brown.

Remove with a slotted spoon, drain on paper towels, and immediately toss with a mix of 1 cup granulated sugar and cinnamon or vanilla powder.

In a saucepan, whisk 3 large egg yolks, 1/2 cup granulated sugar, and 1/8 tsp salt; slowly add 3/4 cup whole milk.

Cook over medium-low heat for about 5 minutes until thickened, whisking continuously.

Remove from heat, stir in 2 Tbsp softened butter, and let cool completely (chill if desired).

In a chilled bowl, whip 1 1/3 cups heavy cream with 1/4 cup powdered sugar to stiff peaks.

Gently fold the whipped cream into the cooled custard until smooth.

Allow the fried donuts to cool completely.

Using a piping bag, carefully fill each donut with the Bavarian cream filling.

Optionally, dust with extra vanilla sugar before serving.