Dissolve yeast in warm water in the bowl of your stand mixer.

Add milk, sugar, salt, eggs, butter, vanilla, and 2 cups of flour.

Beat on low for 30 seconds with the paddle attachment, scraping the bowl if needed.

Beat on medium speed for 2 minutes, scraping bowl occasionally.

Stir in remaining flour until smooth.

Cover and let rise until doubled, about an hour.

Turn the dough onto a floured surface and roll it 1/2-inch thick.

Cut with a floured doughnut cutter.

Cover and let rise again until doubled in size, 30-40 minutes.

Heat vegetable oil in a deep fryer to 350 degrees F and fry until golden brown, about 30 seconds on each side.

Remove from the oil and drain on paper towels for a minute.

Place everything in a saucepan and whisk over a low heat until smooth.

Dip the doughnuts into the glaze and set them on cooling racks over paper-lined baking sheets.