Melt butter in a large skillet over medium heat; saute potatoes, onion, and 1 1/2 teaspoons black pepper until just tender, about 20 minutes. Stir 3 cups milk into potato mixture and heat until almost boiling, about 5 minutes. Remove skillet from heat.

Bring water and chicken bouillon to a boil in a Dutch oven or heavy pot.

Combine flour, 7 tablespoons milk, egg, dill, parsley, 1 teaspoon pepper, and salt together in a bowl until dough is stiff. Add more milk, 1 tablespoon at a time, if needed. Roll dough into ropes about 1/2-inch thick on a work surface. Cut ropes into 1/4-inch pieces and drop into boiling broth. Reduce heat, cover Dutch oven with a lid, and simmer until knoephla begin to float, about 10 minutes.

Stir potato mixture into broth and knoephla; simmer until potatoes are tender, about 20 minutes.