-
Cook onion in butter with 1/4 tsp salt in a heavy medium skillet over medium-low heat, covered, stirring occasionally, 5 minutes.
-
Uncover and cook onion, stirring frequently, until deep golden- brown, about 20 minutes more.
-
Meanwhile, cover potatoes with water in a small saucepan and season well with salt. Bring to a simmer and cook until tender, about 10 minutes. Drain and mash, then stir in onion and cheese.
-
Put a rounded tsp filling in center of a wrapper. Lightly brush wrapper with water, then fold in half (diagonally if square) and press to seal. Form remaining dumplings.
-
Heat oil in a 12-inch nonstick skillet over medium-high heat until hot. Stand dumplings, seam side up, in skillet and fry until bottoms are pale golden, 2 to 3 minutes. Add water, then cover tightly with lid and cook until water has evaporated and bottoms of dumplings are crisp and golden, about 5 minutes.