Combine Dry Ingredients: In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, yeast, sugar, and salt. Mix on low speed to combine the ingredients evenly.

Incorporate Wet Ingredients: Add the eggs and 2/3 cup of cold water. Mix on low until a shaggy dough begins to form. If dry flour remains at the bottom of the bowl, add additional water, 1 teaspoon at a time, until fully incorporated.

Knead the Dough: Replace the paddle attachment with the dough hook. Knead the dough on medium speed for 4 minutes, allowing the gluten structure to begin developing.

Incorporate the Butter: Gradually add the softened butter, two pieces at a time, allowing each addition to be mostly incorporated before adding more. Continue kneading until the dough becomes smooth and pulls away from the sides of the bowl without tearing when gently pressed.

Chill the Dough: Transfer the dough to a lightly greased bowl, cover it with plastic wrap, and refrigerate for a minimum of 8 hours or overnight to develop flavor and texture.

Prepare Baking Sheet: Cut twelve 4x4-inch squares of parchment paper and line a large baking sheet with them.

Roll Out the Dough: Remove the dough from the refrigerator and transfer it to a lightly floured surface. Roll it to a thickness of approximately 1 inch.

Cut the Donuts: Using a 2-inch round cookie cutter, cut circles from the dough. Place each circle on a prepared parchment square.

Proof the Donuts: Cover the donuts loosely with plastic wrap and allow them to proof in a warm location until doubled in size.

Heat the Oil: In a large, deep pan, heat the vegetable oil to 360°F (182°C).

Fry the Donuts: Carefully lower the donuts into the hot oil, parchment paper side up. Once submerged, gently peel away the parchment paper using tongs. Fry each side for 90 seconds or until golden brown.

Drain and Cool: Transfer the fried donuts to a paper towel-lined tray to drain excess oil. Allow them to cool for 5 minutes before coating in granulated sugar.

Heat the Milk: In a medium saucepan over medium heat, bring the milk just to a simmer, avoiding boiling. Once bubbles appear along the edges, remove it from the heat.

Whisk the Yolks: In a separate bowl, whisk together the sugar, flour, and egg yolks until smooth.

Temper the Eggs: Gradually add the warm milk to the egg mixture, one tablespoon at a time, whisking constantly to prevent curdling.

Cook the Custard: Return the mixture to the saucepan and cook over medium-low heat, stirring continuously until the custard thickens enough to coat the back of a spoon.

Finish the Custard: Remove from heat and stir in the vanilla extract and butter until fully incorporated. Transfer the custard to a glass bowl, cover with plastic wrap directly touching the surface to prevent a skin from forming, and refrigerate until fully chilled.

Whip the Cream: In a separate bowl, whip the heavy cream to stiff peaks. Once the custard has cooled, gently fold the whipped cream into the custard until fully combined.

Prepare the Donuts: Roll each donut in granulated sugar or dust with powdered sugar if preferred.

Fill the Piping Bag: Transfer the Bavarian cream into a piping bag fitted with a round nozzle.

Fill the Donuts: Using a small knife, create a hole in the side of each donut and gently fill with the Bavarian cream. Avoid overfilling to prevent leakage.