Start by placing the water and milk in a large microwave-safe bowl and microwave for 1 1/2 minutes or so until warm but not hot or boiling.
Next whisk the sugar into the milk mixture, then sprinkle with the yeast and set aside for 15 minutes until the yeast has become foamy.
Then add the egg, salt, butter, and 2 cups of the flour to the milk mixture. Beat with a wooden spoon or sturdy spatula until the mixture is thick but smooth. Add the remaining 1 1/2 cups flour and beat again until just combined. You’re looking for a thick batter-like consistency.
Dust a work surface with about 1/4 cup more flour and dump the dough onto it. Gently fold the dough over itself 3 to 4 times (the dough will be sticky and a little loose). Gently shape the dough into a round and place in a well-oiled bowl (you can reuse the mixing bowl if it is mostly scraped clean). Cover with a clean kitchen towel.
Set the bowl in a warm place to rise until doubled in size, 1 1/2 to 2 hours. Or you can cover the bowl with plastic wrap and let it rise overnight in the refrigerator for around 8 to 10 hours.
Heat the oil in a large Dutch oven fitted with a deep-fry thermometer over medium-high heat until 350ºF. Line a baking sheet with two layers of paper towels. If the dough is in the refrigerator, pull it out just before you heat the oil to give it 20 to 25 minutes to lose its chill while the oil warms up.
While the oil is heating, dust a work surface with about 1/4 cup of flour and dump the dough onto it. Roll the dough out to a 17x11-inch rectangle that is about 1/2-inch thick. Use your heart shaped cookie cutter (or just a knife to cut into 2 inch squares)
Fry 3 to 4 pieces at a time: Gently add the beignets to the oil using tongs or your hands if you're not a chicken like me. Immediately use a large heatproof spoon to ladle the hot oil over the pieces. Fry the first side until golden brown, 1 to 2 minutes. Using the spoon or tongs, flip the beignets and fry until the second side is golden brown, 1 to 2 minutes more.
Transfer the pieces to a paper towel lined baking sheet and repeat with the remaining dough.
Dust the dough heavily with powdered sugar while they're still warm and serve immediately!