Activate the yeast: In a small bowl, combine the warm milk, yeast, and 1 tablespoon of the sugar. Let stand for 5 minutes, or until the yeast is foamy.
Make the dough: In a large bowl, whisk together 3 cups of the flour and the salt. Add the activated yeast mixture, eggs, melted butter, and vanilla extract. Stir until a dough forms.
Knead the dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or until smooth and elastic.
First rise: Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1 hour, or until doubled in size.
Shape the donuts: Punch down the dough and roll it out on a lightly floured surface to a thickness of about 1/2 inch. Use a 2 1/2-inch round cutter to cut out the donuts, and then use a smaller cutter to cut out the centers.
Second rise: Place the donuts on a baking sheet lined with parchment paper, leaving some space between them. Cover with a clean kitchen towel and let rise for 30 minutes, or until puffy.
Fry the donuts: Heat about 2 inches of vegetable oil in a deep pot to 350°F. Carefully add a few donuts at a time and fry for 1-2 minutes per side, or until golden brown.
Drain and coat: Remove the donuts from the oil with a slotted spoon and drain on paper towels. Immediately roll them in granulated sugar while still warm.
Make the filling: In a chilled bowl, beat the heavy cream with an electric mixer until it forms stiff peaks. Gradually add the powdered sugar and vanilla extract, beating until combined.
Fill the donuts: Use a piping bag fitted with a small round tip to fill the donuts with the cream filling.
