Combine the active sourdough starter, milk, flour, and sugar in a bowl. Cover and let rise in a warm place for 4 hours until the levain has doubled or tripled in size.

Gather all the ingredients. Add everything except the butter to the bowl of a stand mixer fitted with a dough hook. Mix on medium speed until the dough forms. Cover and let it rest (autolyse) for 30 minutes.

Restart the mixer and knead on medium speed for 5-10 minutes.

Gradually incorporate the softened butter, adding one cube at a time after the initial 5-10 minutes of kneading.

Once all the butter is incorporated, continue kneading at medium speed for 15-25 minutes until the dough passes the window-pane test.

Shape the dough into a round and transfer it to a lightly oiled bowl. Proof in a warm spot (around 78°F/26°C) for 2 hours.

After proofing, refrigerate the dough for at least 6 hours, or up to 15 hours, for cold-proofing.

Remove the dough from the fridge and let it rest at room temperature for about 30 minutes. Weigh the dough and divide it into 9 large buns (about 120g each) or 12 smaller buns (about 90g each).

Cut the dough into equal pieces using a bench knife. Shape each piece into a seamless round and place them on a lined baking tray. Gently flatten the rounds into disc shapes. Cover loosely with plastic wrap.

Proof at room temperature (around 68°F/20°C) for 10 hours, until the buns are puffed up and doubled in size. This may take 9-12 hours depending on your environment and starter activity.

Whisk together egg yolks, sugar, corn starch, and salt in a bowl.

Split the vanilla bean pod lengthwise and scrape out the seeds.

In a small pot, heat the milk and vanilla bean seeds over medium heat until small bubbles form around the edges (almost boiling).

Remove from heat and gradually pour the hot milk into the egg mixture, whisking constantly.

Strain the mixture back into the pot using a fine-mesh strainer. Cook over medium heat, whisking continuously until the pastry cream is thick and glossy.

Remove from heat and stir in the butter (add vanilla extract now if using). Transfer to a bowl, cover with plastic wrap touching the surface, and let cool to room temperature before refrigerating.

Preheat the oven to 430°F/220°C.

Make an indention in the center of each bun with your knuckles or fingertips, pressing down firmly to ensure the filling stays in place.

Fill the centers with a generous amount of vanilla pastry cream using a spoon or piping bag.

Brush the buns with egg wash and bake for 15-18 minutes until golden brown. Transfer to a cooling rack.

Once the buns are completely cooled, prepare the glaze.

Mix powdered sugar with water to form a thick glaze.

Drizzle the glaze over each bun with a spoon and sprinkle with desiccated coconut.