In a bowl, combine warm milk and yeast; let it sit for 5 minutes.

In a large mixing bowl, combine flour, sugar, and salt.

Add the yeast mixture, eggs, melted butter, and vanilla to the dry ingredients.

Mix until a dough forms, then knead for about 10 minutes.

Place the dough in a greased bowl, cover, and let rise for 1 hour or until doubled in size.

Punch down the dough and roll it out on a floured surface to about 1/2 inch thick.

Cut out circles using a cookie cutter or glass.

Let the circles rise for an additional 30 minutes.

Heat vegetable oil in a deep fryer or pot to 350°F (175°C).

Fry the doughnuts until golden brown, about 2-3 minutes per side. Drain on paper towels and fill with your choice of filling.