Prep your potato mixture by cooking a potato and mashing it with butter. Mix flour and egg with enough water to make a soft dough. Knead well on a flat surface.
Roll out on a floured cutting board or counter until the dough becomes thin.
Cut the dough into squares.
Place 1 teaspoon of potato filling on each square. Fold in half to make a triangle. Pinch the edges well to secure the filling inside.
Drop each triangle into boiling salted water and cook until they rise to the surface. Cook them for five minutes longer.
Remove from the water and place in a colander. Rinse them with hot water. Drain and pour melted butter over the pirohy and serve (with grated cheese if you want!).