Step 1: Make the Custard Filling

1. Heat the Milk: In a medium saucepan, heat the milk over medium heat until it begins to steam but does not boil.
2. Whisk the Yolks: In a separate bowl, whisk together the egg yolks, sugar, and cornstarch until smooth.
3. Temper the Eggs: Slowly pour half of the hot milk into the egg mixture while whisking constantly. This step prevents the eggs from curdling.
4. Cook the Custard: Return the tempered mixture to the saucepan and cook over medium heat, whisking continuously, until the custard thickens and begins to bubble.
5. Finish with Butter and Vanilla: Remove from heat and stir in the butter and vanilla extract. Transfer the custard to a bowl, cover with plastic wrap (pressed directly onto the surface), and refrigerate for at least 2 hours until chilled.

Step 2: Prepare the Doughnut Dough

1. Activate the Yeast: In a small bowl, mix the warm milk with a pinch of sugar and sprinkle the yeast on top. Let it sit for 5-10 minutes until foamy.
2. Mix the Dough: In a large mixing bowl, combine the flour, sugar, and salt. Add the yeast mixture, eggs, and softened butter. Mix until a sticky 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. First Rise: Place the dough in a greased bowl, cover with a damp towel, and let it rise in a warm place for 1 hour, or until doubled in size.

Step 3: Shape and Fry the Doughnuts

1. Roll and Cut: Punch down the risen dough and roll it out to about 1/2-inch thickness. Use a 3-inch round cutter to cut out circles. Re-roll scraps as needed.
2. Second Rise: Place the dough rounds on a parchment-lined baking sheet, cover, and let rise for 30 minutes.
3. Heat the Oil: Heat about 2 inches of vegetable oil in a large pot or deep skillet to 350°F (175°C).
4. Fry the Doughnuts: Fry the doughnuts in batches, about 1-2 minutes per side, until golden brown. Transfer to a wire rack lined with paper towels to drain excess oil.

Step 4: Fill the Doughnuts

1. Prepare the Filling: Transfer the chilled custard to a piping bag fitted with a long nozzle or round tip.
2. Poke and Fill: Use a skewer or knife to make a small hole in the side of each doughnut. Insert the piping tip and fill each doughnut with custard until slightly puffed.

Step 5: Add the Caramelized Sugar Topping

1. Sprinkle with Sugar: Dip the tops of the filled doughnuts in granulated sugar or sprinkle a thin, even layer on top.
2. Caramelize with Torch: Use a kitchen torch to caramelize the sugar until golden brown and crackly. Allow to cool slightly before serving.