Combine warm water and yeast in a bowl and let sit for 10 minutes until foamy.

Stir in 1 cup of flour and let rise in a warm place for 40 minutes.

Beat butter and sugar until light and fluffy.

Add eggs one at a time, then vanilla and salt.

Add yeast mixture and remaining flour, mix until dough forms.

Place dough in a greased bowl, cover, and let rise for 1 1/2 hours.

Knead the dough slightly and wrap tightly in plastic wrap.

Refrigerate for at least 3 hours, or overnight.

Roll out dough to 1/4 to 1/2 inch thickness on a floured surface.

Cut out donuts using a floured donut cutter.

Place donuts on waxed paper and let rise for 30 minutes until puffy.

Heat vegetable oil to 375 degrees.

Fry donuts, 3 at a time, until browned on both sides (about 1 1/2 minutes).

Drain donuts on paper towels.

Whisk together confectioners' sugar, milk, and corn syrup for glaze.

Microwave glaze for 40 seconds to warm.

Brush glaze over warm donuts.

Cool slightly and serve.