Make the dough: Place the flour and salt into a mixing bowl and stir to combine. Add the egg and half of the water, and mix with a wooden spoon. Once the water is incorporated add the remaining water and mix until a dough comes together.

On a heavily floured surface, use your hands to knead the dough until the it is smooth and homogenous. Split the dough in two equal pieces. Roll each piece into a ball, and cover each half in plastic wrap. Set aside at room temperature for at least 2 hours.

Add the grated onion, salt, and pepper to a large mixing bowl, and stir to combine. Set aside for about 10-15 minutes. Then add the chopped beef to the onion mixture and stir to combine. Cover and refrigerate.

Transfer the rested dough onto a heavily floured surface. Knead the dough for 2-4 more minutes until it is no longer sticky. Use a rolling pin to roll out the dough into a thin circle 15 inches in diameter. Sprinkle the dough with flour and fold the top edge over the bottom to create a semi circle, and then fold the semi circle into a quarter of a circle. Loosely cover the dough with plastic wrap and set it aside to rest for 5 minutes.

Place the dough onto a floured surface and unfold the dough back into a circle. Sprinkle the dough with flour. Roll out the dough as thinly as you can, until the circle has about doubled its size. Cut out 4-inch circles with a circular cookie cutter. 

Fill and shape the dumplings: Fill a circle of dough with about 3½ tablespoons (or as much as can fit) of the beef filling. Pick up the piece of dough with filling into the palm of your hand. Using your other hand, crimp together the two halves of the circle over the filling to create a stuffed half moon shape. Repeat this step with the other dough circles until you use all of the filling. Transfer the manti to the freezer for 20 minutes to set.

Meanwhile, pour about 2 inches of water into a pot and then place an oiled steamer or flat colander into the pot. Carefully place the dumplings into the steamer or colander, making sure to leave space between the dumplings in the steamer. Put a lid on the pot and steam the dumplings over simmering water until the filling is cooked through and the dumplings are tender, about 30-45 minutes. If needed, steam the dumplings in two batches. 

Serve immediately. Leftover manti can be stored in an airtight container in the refrigerator for 4 days.