In a stand mixer bowl, combine 2 cups of flour, yeast, sugar, and salt. Stir them well.

Gently heat the milk and butter on low heat until the butter melts, reaching about 110-115F. Pour this warm mixture over the dry ingredients and mix until just combined.

Lightly scramble the eggs and mix in the vanilla. Add this mixture to the stand mixer and blend until smooth.

Switch to a dough hook and gradually add the remaining flour. Keep mixing until a soft dough forms that pulls away from the bowl but remains sticky.

Place the dough in an oiled bowl. Cover it and let it rise in a warm area until it doubles in size, approximately 30 minutes.

Roll out the dough to about ½ inch thickness. Use a donut cutter to cut into shapes. Let the donuts rise again for another 30 minutes.

Heat oil to 350-355F in a deep saucepan. Fry the donuts for about 60 seconds on each side until they are golden brown.

Immediately dust with sugar or dip them in glaze. For the glaze, whisk together powdered sugar, melted butter, vanilla, and hot water until smooth. Let donuts cool for the glaze to harden before serving.