Pour the warm water into a large mixing bowl or the bowl of a stand mixer. Stir in the active dry yeast and 1 teaspoon of the granulated sugar. Let the mixture sit undisturbed for 5 to 10 minutes until it becomes frothy and develops a foamy layer on top.

Add the whole milk, the remaining granulated sugar, salt, slightly cooled melted butter, and the egg to the yeast mixture. Whisk everything together thoroughly until the egg is completely broken down and combined.

Stir in 3 cups of the all-purpose flour using a sturdy spoon or a stand mixer fitted with a dough hook until a dough begins to form. Gradually add the remaining 1/2 cup of flour. Turn the dough out onto a lightly floured surface and knead by hand for 3 to 5 minutes, or mix on medium-low speed in your mixer, until the dough is smooth and slightly tacky.

Shape the dough into a ball and place it inside a lightly oiled bowl, turning it over once to grease the top. Cover the bowl tightly with plastic wrap and leave it in a warm, draft-free area to rise for 1 1/2 to 2 hours, or until it has doubled in size.

Gently punch down the dough to deflate it and transfer it onto a well-floured work surface. Roll the dough out with a rolling pin into a large rectangle that is roughly 1/4-inch thick. Cut the dough into 2 1/2-inch squares using a sharp knife or a pizza cutter, then let the squares rest on the counter for 10 minutes.

Pour at least 2 inches of vegetable oil into a heavy-bottomed pot or Dutch oven and set it over medium heat. Bring the oil to 370°F, checking the temperature with a candy or deep-fry thermometer.

Carefully slide 3 or 4 dough squares into the hot oil, working in batches so you do not crowd the pot. Fry for 1 to 2 minutes until the bottoms are puffed and golden brown, then flip them over and fry for another 1 minute on the other side.

Remove the beignets from the oil using a slotted spoon or spider strainer and place them immediately onto a wire rack lined with paper towels to drain. Transfer the hot pastries to a plate and dust them with a heavy layer of powdered sugar before serving. Enjoy!