Step 1: Make the Custard

1. Heat the Milk: In a medium saucepan, heat the milk over medium heat until it begins to steam. Do not let it boil.
2. Whisk the Egg Yolks: In a separate bowl, whisk together the egg yolks, sugar, and cornstarch until smooth.
3. Temper the Eggs: Slowly pour half of the warm milk into the egg mixture while whisking constantly to prevent curdling.
4. Cook the Custard: Return the tempered egg mixture to the saucepan with the remaining milk. Cook over medium heat, whisking constantly, until the mixture thickens and begins to bubble.
5. Add Butter and Vanilla: Remove from heat and stir in the butter and vanilla extract. Transfer to a bowl, cover with plastic wrap (pressed directly onto the surface), and refrigerate until completely cool.

Step 2: Prepare the Dough

1. Activate the Yeast: In a small bowl, mix the warm milk, sugar, and yeast. Let it sit for 5-10 minutes until frothy.
2. Mix the Dough: In a large mixing bowl, combine the flour and salt. Add the yeast mixture, melted butter, and egg. Mix until a shaggy dough forms.
3. Knead the Dough: Turn the dough onto a floured surface and knead for 8-10 minutes until smooth and elastic. Alternatively, use a stand mixer with a dough hook for 5-7 minutes.
4. Let It Rise: Place the dough in a lightly greased bowl, cover with a clean kitchen towel, and let it rise in a warm place for 1 hour, or until doubled in size.

Step 3: Assemble the Buns

1. Divide the Dough: Punch down the risen dough and divide it into 8 equal pieces. Roll each piece into a ball.
2. Flatten and Fill: Flatten each dough ball into a 4-inch circle. Place a spoonful of chilled custard in the center of each circle. Gather the edges of the dough and pinch them together to seal, forming a smooth bun.
3. Rest the Buns: Place the buns seam-side down on a parchment-lined tray. Cover with a towel and let them rest for 15 minutes.

Step 4: Cook the Buns

1. Heat the Pan: Heat a large, nonstick skillet over low to medium-low heat. Lightly grease with vegetable oil or butter.
2. Cook the Buns: Place the buns in the skillet, leaving space between them. Cover the skillet with a lid and cook for 8-10 minutes on the first side, or until golden brown. Flip the buns and cook for another 6-8 minutes on the other side.
3. Check for Doneness: The buns should sound hollow when tapped on the bottom. Adjust the heat as needed to prevent burning.

Step 5: Serve and Enjoy

1. Let the buns cool slightly before serving. Dust with powdered sugar or brush with honey for added sweetness. Enjoy warm or at room temperature.