In a large bowl, whisk together the flour, granulated sugar, salt, and active dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix the ingredients until combined, then knead the dough for 8-10 minutes until it becomes smooth and elastic.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
Meanwhile, prepare the filling: in a bowl, beat together the cream cheese, melted butter, powdered sugar, and cornstarch until you have a silky, smooth cream.
On a lightly floured surface, roll out the dough to about 1 cm (1/3 inch) thickness. Use a glass or doughnut cutter to cut out circles.
Place a small amount of the buttery filling in the center of each dough circle, then fold it in half and seal the edges tightly to prevent the filling from leaking out during frying.
Place the filled doughnuts on a baking sheet lined with parchment paper, and cover them with a clean kitchen towel. Let them rise for another 30 minutes.
In a large pot, heat the oil to approximately 340°F (170°C). Test the oil by dropping in a doughnut: if it immediately floats to the surface and gently sizzles, the oil is ready.
Fry the filled doughnuts for 2-3 minutes per side, until golden brown. Use a slotted spoon to remove them, then place them on paper towels to absorb excess oil.
Sprinkle the finished doughnuts with powdered sugar, and serve fresh while the filling remains soft and creamy.
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