Add flour and sugar to a bowl of a stand mixer fitted with a dough hook. Feel free to use a hand-held electric mixer fitted with two dough hooks. Make a well in the middle. Pour lukewarm milk and lukewarm water into the well. Add the active dry yeast and cover the ingredients with some flour. Add the egg and salt. Start kneading on low, set the stand mixer to medium speed, and knead for about 6 - 8 minutes or until the dough is elastic, smooth, and easily separates from the bowl. Add the butter and knead for another 3 - 4 minutes for the butter to incorporate into the dough and separates from the bowl easily.TipThe water and milk temperature should be around 30 - 35°C or 85 - 95°F.
Cover the bowl with the dough with clingfilm and leave to rise at room temperature (20 - 25°C / 68 - 77°F) until visibly increased, for about 1 hour - 1 1/2 hours. Then place the dough in the fridge for 30 - 60 minutes. Feel free to leave the dough in the fridge overnight.
Generously dust your working surface with flour. Dust your hands as well. Spread the dough to get a rectangle, then roll it into 0.5 cm / 0.2-inch thickness using a rolling pin. Cut into 18 squares. Place a large pan (with a lid) over medium heat. Add the oil and wait until the oil reaches 170 - 180°C / 340 - 355 °F. The beignets are airy and soft, so carefully pick each and shake off any excess flour. Drop them into hot oil and wait for each to float on top. Then carefully dip them in oil for a few seconds using a fork and wait for them to puff up. Fry for 2 - 3 minutes, then turn them and fry for another 2 - 3 minutes. Transfer the beignets to a baking sheet lined with paper towels and repeat the process with the rest of the dough.
Generously sprinkle the Beignets with powdered sugar, then serve them as soon as possible.