Pour the milk and butter into a medium saucepan and heat over medium heat until the butter is melted. Remove from heat and set aside.
In a small bowl, sprinkle the yeast over the warm water and let it dissolve for 5 minutes.
The mixture should become frothy.
After 5 minutes, pour the yeast mixture into the stand mixer bowl, fitted with the paddle attachment, and add the milk and butter mixture.
Add the eggs, sugar, salt, vanilla, and half of the flour.
Combine the ingredients at low speed until the flour is incorporated, then increase the speed to medium and beat until smooth.
Add the remaining flour, at low speed, then increase the speed to medium and beat well.
Change the paddle for the dough hook and beat at medium speed until the dough detaches from the bowl and becomes smooth, about 3 to 4 minutes.
Transfer to a well-oiled bowl, cover, and let rise for 1 hour or until the dough has doubled in volume.

On a well-floured surface, roll out the dough to 3/8″ (9 mm) thickness.
Cut out the dough using a beignet cutter 2 and 1/2″ (65 mm) in diameter.
Place the beignets on a floured baking sheet, cover lightly with a towel, and let rise for 30 minutes.

Preheat the oil in a deep fryer or a large pot to 365 F/185 C.

Gently place the beignets in the oil, 3 to 4 at a time.
Cook for 1 minute on each side or until golden brown.
Drain on paper towels.
Wait 15 to 20 minutes before glazing.