Using your electric mixer with the dough hook attachment, first mix the flour, egg, salt, baking powder, milk, and softened butter together on the lowest setting. Keeping the speed at “stir,” slowly add the water in a few separate batches (you may need more or less depending on the humidity in your kitchen).
Knead the dough for 15 minutes. The dough should be sticky, but not stick to the sides of the bowl. Cover the dough, and let rest for 30 minutes. This can be done by hand if you don’t have a mixer. Just knead the dough for 5-10 minutes longer.
On a clean, lightly floured surface, form the dough into a long flat loaf shape, about 1/4-inch thick and 4 inches wide. Take the time to make it truly uniform. Place it in the center of a large piece of plastic wrap on a baking sheet or long, flat plate. Wrap the dough, tucking the two ends of the plastic under the loaf, and ensuring that the dough is completely covered. Refrigerate overnight.
In the morning, take out the dough and let it sit on the counter (wrapped) for 1-3 hours until the dough is completely back to room temperature and VERY, VERY soft to the touch. This step is critical, and may take longer in the colder months. If you don't let the dough come back to room temperature, it won't fry up properly.
Now prepare the oil for frying using your wok. You can also use a large pan with some depth for added safety. The goal is to have a large vessel, so that you can produce authentically long crullers. Use medium heat to slowly bring the oil up to 400-425°F/205-220°C.
While the oil is heating up, you can unwrap the dough. Gently flip the dough onto a lightly floured surface, peeling off the plastic wrap. Very lightly flour the top side of the dough also. Next, cut the dough into 1-inch wide strips (try to cut an even number of strips). Then stack them two by two, and press the center, lengthwise, with a chopstick. Hold the two ends of each piece, and gently stretch the dough to a 9-inch long rope.
Once the oil is up to temperature, carefully lower the stretched dough into the oil. If the oil temperature is right, the dough should surface right away. Now take a long cooking tool (we used chopsticks, but you could also use tongs), and quickly roll the dough in a continuous motion for about a minute.
You can fry one to two at a time. Just be sure to take the time to continuously roll the dough in the oil. The youtiao is done once they turn light golden brown. Try not to over-fry them as they become unpleasantly crunchy rather than chewy and delicious.
Now, repeat those steps with the remaining dough. You might want two people manning the process—one to form and place the dough into the fryer and one to roll the dough around once it’s in the oil.