Make the dough: Mix the flour and salt in a large mixing bowl. Add the egg yolks and water and mix well with a fork until combined. Turn out the dough and knead on a work surface until a smooth dough is formed, about 5 minutes. Place in an oiled bowl and cover with plastic wrap and let rest for at least 15 minutes or up to 1 hour at room temperature.

Make the filling: Grate the onion on the large holes of a box grater. Add to a medium mixing bowl, along with the beef, oil, pepper, and salt, mixing well to combine. Cover and refrigerate.

Form the khinkali: Divide the dough into 2 portions. Roll each portion out on a lightly floured surface as thin as possible (about 1/16-inch thick). Use a 2½-inch round cutter or small glass to punch out 30-35 circles, re-rolling dough scraps as needed. Add ½ tablespoon of the filling to the center of each circle. Fold in half and pinch the edges with your fingers to seal. Transfer the finished khinkali to a parchment-lined sheet tray, cover with plastic wrap, and repeat with remaining dough. 

Bring the stock to a rolling boil in a large pot, and add the khinkali in 2 batches to avoid them sticking together. Reduce to a simmer and cook the khinkali until the dough and meat are no longer raw and the dough feels more light and airy rather than chewy, about 10 minutes.

To serve, add 2 ladles of hot broth and khinkali to a bowl. Place a small bowl of broth with the minced garlic on the table to add to the bowls, as desired.