Heat 4 tablespoons of the oil in a medium pan over medium-high heat. Season the chicken and liver pieces lightly with salt. Add to the pan and sear, turning occasionally, until browned on all sides and cooked through, 5 to 7 minutes. Transfer the pieces to a plate and set aside to cool slightly.

Add the remaining oil to the pan along with the onions and sauté, stirring occasionally, until the onions are golden brown and beginning to caramelize, about 10 minutes. Season lightly with salt and remove from the heat. Set aside to cool slightly.

Transfer the chicken and liver to a food processor, along with the onions, 1½ tablespoons of the beaten egg, the ½ teaspoon of the salt, and all of the pepper. Pulse the mixture until it is finely ground and reseason to taste. Transfer to a small bowl and cover loosely with plastic wrap. Set aside to cool completely.

Meanwhile, make the dough: Combine the flour and salt in a medium bowl and make a well in the center. Whisk the egg with 1 cup of water and the remaining egg from the filling. Pour the egg mixture into the well in the dry ingredients and mix with a fork until a shaggy dough forms.

Turn the dough onto a lightly floured surface and knead until soft and smooth, 1 to 2 minutes. Return the dough to the bowl, cover with a towel or plastic wrap, and set aside to rest for 20 minutes.

Line a baking sheet with wax or parchment paper and drizzle evenly with oil. Divide the dough into thirds with a dough cutter or chef’s knife. Transfer one-third of the dough to a heavily floured surface, keeping the other two-thirds covered. Using a rolling pin, roll out the dough as thinly as possible into an 20-inch by 14-inch rectangle.

Using a pizza cutter or chef’s knife, cut the rolled dough into 2-inch squares. Place ½ teaspoon of filling in the middle of each square. Lightly brush the edges with water and fold each square in half over the filling to form a triangle, pressing the edges gently together to create a tight seal. Fold the two longer corners inward to connect and clasp along the longest edge of each dumpling, pressing to seal tortellini-style. Place on the prepared baking sheet, cover loosely with plastic wrap, and continue with the remainder of the dough and filling.

While forming the dumplings, bring a large pot of generously salted water to a boil. Working in two batches, gently slide the dumplings into the water. Simmer 3 to 4 minutes until the dumplings float to the surface. Remove the dumplings with a slotted spoon to a wax or parchment paper-lined tray.

To serve, add the cooked dumplings to hot chicken soup or fry them in a pan with onions.