Hello Laura! First of all, congratulations on making khinkali from scratch. Getting that rich, flavorful broth sealed properly inside the dough is an achievement in itself, so you should be very proud of the effort.
Spilling the broth is actually the most common first-timer mistake, and it usually happens because cutlery gets involved. The primary rule of eating authentic khinkali is to never use a fork and knife. Puncturing the dough with utensils allows all of that delicious, seasoned soup to drain onto the plate instead of being savored.
Here is the traditional, step-by-step method to eat them cleanly with your hands:
1. Let the khinkali rest for a couple of minutes after boiling so the broth is warm rather than scalding, and dust them lightly with freshly cracked black pepper if you enjoy a little extra heat.
2. Grab the dumpling firmly by its pleated top knob, known in Georgia as the kudi or handle. This part is meant specifically to be held by your fingers.
3. Carefully flip or tilt the dumpling so the broad base faces upward or toward you, and take a small, gentle bite into the side of the dough.
4. Bring the opening to your lips and carefully slurp out all of the warm meat broth before taking another bite.
5. Once the soup has been enjoyed, eat the rest of the tender dough and spiced meat filling.
Traditionally, the thick dough stem you were holding is left on the edge of the plate. Because it is quite dense and exists mostly as a handle, it is rarely eaten, and diners often leave their stems in a neat pile to see who ate the most at the table.
I hope this helps you enjoy your next batch to the fullest!