In a large bowl, combine warm milk, sugar, and yeast; let it sit for 5-10 minutes until foamy. Add eggs, melted butter, vanilla, and salt, then gradually mix in the flour until a shaggy dough forms.

Knead the dough on a lightly floured surface for 5-7 minutes until smooth and elastic. Place the dough in a greased bowl, cover, and let it rise in a warm place for 1-1.5 hours, or until doubled in size.

While the dough rises, whisk together milk, sugar, cornstarch, and egg yolks in a saucepan. Cook over medium heat, stirring constantly, until the mixture thickens; remove from heat and stir in butter and vanilla extract.

Transfer the cream filling to a bowl, cover with plastic wrap directly on the surface to prevent a skin from forming, and chill completely in the refrigerator for at least 2 hours.

Once the dough has risen, punch it down and roll it out on a lightly floured surface to about 1/2-inch thickness. Cut out circles using a 3-inch round cutter.

Place the cut donut circles on parchment-lined baking sheets, cover loosely, and let rise for another 30-45 minutes until puffed.

Heat vegetable oil in a deep pot or Dutch oven to 350-360°F (175-180°C). Carefully fry donuts 2-3 at a time for 1-2 minutes per side until golden brown, then transfer to a wire rack lined with paper towels to drain.

Once cooled, use a small knife or skewer to create a pocket in the side of each donut. Fill a piping bag fitted with a bismarck or round tip with the chilled vanilla cream and pipe it into each donut until plump.

Dust the filled donuts generously with powdered sugar before serving.