In a medium-size bowl, add the warm water, sugar, and yeast and whisk well to combine. Sit aside for about 10 minutes, or until the mixture has bubbled up and become foamy. In the bowl of a stand mixer fitted with the paddle attachment, beat the eggs until smooth. Beat in the vanilla and evaporated milk. Beat in 3 and 1/2 cups of the flour until smooth.
Turn the mixer to low speed and slowly pour in the yeast mixture (careful here - this mixture can splash up if added to quickly!); beat until smooth. Add in the butter and beat until incorporated. Finally, beat in the remaining 3 and 1/2 cups of flour and salt. Beat until dough is smooth and cohesive; about 2 minutes. Cover the bowl tightly with plastic wrap and refrigerate the dough for at least 2 hours, or up to 24 hours.
Line a large rimmed baking sheet with three layers of paper towels, set aside.
In a large enameled cast-iron pan, heat 4-inches of oil to 360 degrees (F). Remove the dough from the refrigerator. On a lightly floured surface, roll the dough out into a 1/4-inch thick rectangle. Cut the dough into 2 and 1/2-inch squares.
In batches, fry the dough until they puff up and are golden brown in color, about 1 minute. Using a slotted spoon, transfer the beignets to the prepared baking sheet and repeat with remaining dough.
Dust with powdered sugar and serve at once!