In a large mixing bowl, whisk together flour, sugar, yeast, and salt. In another bowl, combine warm milk, melted butter, and egg. Mix wet into dry to form a sticky dough.

Lightly oil a bowl, place dough in it, cover, and let rise for about 1-1.5 hours until doubled.

Punch down dough, roll out to 1/2-inch thickness, and cut into donuts. Place on a floured sheet and let rise for 30 minutes.

Heat oil to 350°F. Fry donuts for 1-2 minutes per side until golden brown. Drain on paper towels.

Whip together heavy cream, powdered sugar, and vanilla until stiff peaks form. Transfer to piping bag.

Once cooled, fill donuts with cream using a piping bag.