In a medium mixing bowl, combine the flour, egg, and salt. Mix until the ingredients form a stiff dough. If the dough is too dry, you can add a small amount of water, a teaspoon at a time, until the dough comes together.
On a floured surface, roll the dough into thin ropes about 1/4 inch in diameter.
Cut the ropes into small pieces about 1/4 inch long. You can also pinch off small pieces of dough to form irregularly shaped dumplings if you prefer.
In a large pot, bring the chicken or vegetable broth to a boil.
Add the dumplings to the boiling broth, stirring gently to prevent them from sticking together.
Reduce the heat to a simmer and cook the dumplings for about 10-15 minutes, or until they are cooked through and tender.
Stir in the butter and let it melt into the broth.
Season the soup with additional salt to taste, if necessary.
Ladle the zacierki into serving bowls.
Garnish with freshly chopped dill or parsley.
Enjoy your meal!