Pour the milk into a medium-sized saucepan and place it over medium heat. Cook until it starts to steam and the edges just begin to bubble, then remove from heat.

In a medium bowl, whisk together the egg yolks, sugar, and cornstarch until smooth. Slowly pour the hot milk over the egg mixture, whisking continuously to gradually warm the eggs.

Strain the mixture back into the saucepan and place it back over medium heat. Whisk constantly until the mixture comes to a boil and thickens to a soft pudding-like consistency, which should take about 5 to 10 minutes.

Remove the saucepan from the heat and whisk in the butter and vanilla extract. Pour the pastry cream into a heatproof bowl and cover it with plastic wrap, ensuring the wrap touches the surface of the cream to prevent a skin from forming.

Refrigerate the pastry cream for 1 to 2 hours, or until it’s chilled and set.

Warm the milk over low heat on the stove or microwave it for about 30 seconds until it feels warm to the touch, but not hot.

In a large bowl, whisk together the warm milk, 1 tablespoon of sugar, and the yeast. Set the mixture aside for 5 to 10 minutes, or until the surface starts to bubble and foam. If it doesn’t foam, the yeast may be expired or the milk might have been too hot, and you'll need to start again.

Add the remaining sugar, egg, vanilla extract, and melted butter to the bowl. Whisk until combined.

Gradually add half of the flour to the wet ingredients. Use a spatula or wooden spoon to mix until it forms a rough, sticky dough. Add the remaining flour and salt, then mix until it comes together into a ball of dough.

Turn the dough out onto a floured surface and knead for 5 to 10 minutes, adding flour as needed to prevent sticking, until the dough becomes smooth and elastic.

Grease a large bowl with oil and place the dough in it, turning it to coat the dough in oil. Cover the bowl and set it aside in a warm place to rise for 45 minutes to 1 hour, or until the dough has doubled in size.

Line two large baking sheets with parchment paper and set them aside.

Once the dough has risen, punch it down and turn it out onto a lightly floured surface. Roll the dough out to about ½ inch thick, and use a 3-inch round cookie cutter to cut out donut shapes.

Place the cut donuts on the parchment-lined baking sheets, leaving at least 3 inches between each one. Gather any leftover dough, reroll it, and continue cutting out donuts.

Cover the cutout donuts and let them rise in a warm place for 30 to 45 minutes, or until they have doubled in size.

Preheat the oven to 375°F (190°C).

Once the donuts have risen, uncover them and bake in the center of the preheated oven for 10 to 12 minutes, or until they are puffed up and golden brown on top. Be sure not to overbake, or they will become dry.

Remove the donuts from the oven and transfer them to a wire rack. Let them cool completely before filling.

To fill the donuts, use a sharp paring knife to make a hole in the side of each one. Fill a large piping bag with the chilled pastry cream and cut the tip off. Insert the tip into the hole in each donut and pipe the pastry cream in until it starts to push out.

Once filled, place the donuts in the fridge until you're ready to top them with caramelized sugar. If you’re not serving them immediately, it’s best to wait until closer to serving to make the sugar topping to keep it crisp.

In a medium skillet, combine the sugar, water, and salt. Set the skillet over medium-high heat and bring the mixture to a boil. Gently swirl the pan if necessary to ensure the sugar melts evenly. Avoid stirring, as it can cause the sugar to crystallize.

Continue boiling the sugar until it begins to brown. As soon as it starts to turn honey-colored, remove the pan from the heat. The sugar will continue to brown slightly from the residual heat, so it’s best to take it off a little earlier to avoid burning.

Let the sugar sit for about 1 minute off the heat to reach an amber color. If it hasn’t browned enough, return the pan to the heat for a few seconds. Be careful not to overcook it, as the sugar will taste burnt.

Using long-handled tongs, hold each donut and dip the tops into the caramelized sugar. If the sugar hardens too much during the process, warm it over low heat to liquefy it again.

Place the caramelized donuts on a wire rack or a piece of parchment paper to catch any drips. Let the sugar harden, which should take about 5 to 10 minutes. Once set, serve the donuts and enjoy!