Place water, butter, sugar, and salt in a saucepan over medium-high heat. Stir until the butter is melted and the mixture comes to a boil. Then, reduce the heat to medium.
Add the flour to the mixture and stir to combine well with the butter mixture. Continue to stir and cook until the dough pulls away from the sides and starts to form a ball, about 1 minute. Let it cool for 5 minutes.
Using a hand mixer or stand mixer, beat the dough with the eggs, starting with one egg until it is completely incorporated. Add the remaining eggs one at a time. Chill the dough in the refrigerator for one hour.
After chilling, shape the dough into small balls using a teaspoon or a 1/2 tablespoon. Arrange the dough balls on a plate lined with parchment paper.
Heat 2-3 inches (5 cm to 7 cm) of oil in a frying pan over medium-low heat. Once the oil is fully heated, drop the dough balls in batches and fry them. Keep turning the dough balls to ensure they brown evenly and puff up considerably.
The beignets are ready when they begin to form a seam, indicating that they are bursting. Use a strainer to transfer them out and drain on paper towels. Sprinkle generously with powdered sugar and serve warm.