In a large bowl, whisk together the flour, sugar, salt, and dry yeast.
Add the warm milk, eggs, and vanilla extract, then begin to mix the dough until it comes together.
Melt the butter and gradually add it to the dough, kneading continuously until you get a smooth, elastic dough.
Cover the dough with a clean kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1/2 inch (1.5 cm) thickness. Use a large cookie cutter or glass to cut out circles.
In a deep pot or fryer, heat the oil to 350°F (180°C). Test the oil temperature with a small piece of dough – if it immediately floats to the top and gently fries, the oil is ready.
Fry the doughnuts for 2-3 minutes on each side, until golden brown. Do not overcrowd the pot, to prevent the oil temperature from dropping.
Remove the fried doughnuts and place them on a paper towel-lined plate to absorb any excess oil, then let them cool completely.
Whip the cold heavy cream until stiff peaks form, then transfer it to a piping bag fitted with a nozzle.
Make a small hole on the side of each doughnut and use the piping bag to fill the doughnuts with whipped cream.
Dust the tops with powdered sugar and serve immediately.
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