Dissolve the yeast in the warm water. Add one tablespoon of the sugar. Allow the yeast to sit for 10 minutes to activate.
Heat milk and butter in a small saucepan over medium-low heat until the butter is melted and the temperature reaches 120 degrees with a candy thermometer.
Combine the flour and remaining two tablespoons of sugar.
Add the yeast mixture and warmed milk to the flour and mix for two minutes using a dough hook.
Add the egg yolk and vanilla extract. Continue to whisk for another minute. Dough should pull away from the sides of the bowl and be sticky, but not wet. If it's wet, add a little flour. If the dough doesn't come together, add a little more warmed milk.
Lightly coat the top of the dough with oil. Cover the dough with a towel and allow it to rise in a warm place for one hour.
Heat 2-3 inches of oil over medium heat until it reaches 375 degrees.
In the meantime, turn the dough out onto a flour cutting board or workspace.
Roll the dough out to 1/4 inch thickness. Cut the dough into 2 x 3-inch strips.
Place 3-4 dough strips at a time into the hot oil. Fry for 45 seconds. Carefully spoon some hot oil over the top of the dough.
Turn the beignets over and fry for another 45 seconds.
Strain the beignets from the hot oil and place them on a layer of paper towels to drain.
Serve hot with powdered sugar.