In a large mixing bowl, combine the grated potatoes, grated onion, beaten egg, flour, salt, and black pepper. Mix well until all ingredients are thoroughly combined.
Using a spoon or your hands, shape the potato mixture into small dumplings, about the size of a walnut.
In a large pot, bring the chicken or vegetable broth to a gentle boil.
Carefully drop the potato dumplings into the boiling broth.
Reduce the heat to medium-low and simmer the dumplings for about 15-20 minutes, or until they are cooked through and float to the surface.
In a small skillet, melt the butter over medium heat until it becomes golden brown and fragrant.
Drizzle the browned butter over the cooked dumplings.
Ladle the kletski into serving bowls, ensuring each bowl gets some broth and dumplings.
Garnish with freshly chopped dill or parsley.
Enjoy your meal!