In a large mixing bowl, using the whisk attachment, combine water, sugar, and yeast and stir to combine. Let sit for about 10 minutes until it starts to get foamy on the surface.
Preheat oven to 200°F. Line 3 baking sheets with parchment paper. Set aside.
To the mixing bowl add egg, buttermilk, 3 tbsp butter, salt and vanilla and whisk to combine.
If available, replace whisk attachment with a dough hook. Slowly add in the flour and stir on medium-low speed until the dough begins to remove itself from the sides of the bowl. If needed add in additional flour in 1 Tbsp. increments until the dough pulls itself away from the bowl.
Knead dough on medium speed for 5 minutes.
Transfer dough to a lightly greased bowl. Turn off preheated oven and place bowl in the oven with the door slightly ajar to rise for 15 minutes. *Cover bowl with a towel that has no danger of touching the heating elements of the oven.
Remove bowl from the oven, close the door and preheat to 200°F.
On a lightly floured surface roll dough out into a rectangle or square, about 1/4 inch thick.
Cut 2-inch squares with a pizza cutter.
Place dough on prepared baking sheets. Turn off preheated oven and place the baking sheets in the oven with the door slightly ajar to rise for 30 minutes. *Cover the baking sheets with plastic wrap that has been sprayed with baking spray.
Remove baking sheets from oven, close the door and preheat to 375°F. Brush the tops lightly with milk.
Bake individual baking sheets for 10-12 minutes until light golden brown. Let cool for about 5 minutes then brush the beignets tops lightly with 1 Tbsp. melted butter and generously dust with powdered sugar.
Beignets are best eaten warm and fresh.