For the Schlutzkrapfen, mix flour with egg, oil, salt and about 500 ml of water to make a smooth dough. Allow to rest. Roll out thinly on a floured work surface and cut out slices with a round cutter (approx. 8 cm Ø).
Finely chop onion and fry in butter until golden brown. Add chopped parsley, stir in flour and pour in milk. Mix in strained spinach and season with salt, pepper and nutmeg.
Place some filling on each dough slice, lightly brush the edges with water and fold dough over each other in a crescent shape. Press edge down well. In a large pot, bring enough salted water to a boil and cook the Schlutzkrapfen in it for about 10 minutes.
Lift out with a slotted spoon, drain and arrange on preheated plates. Sprinkle with grated cheese and drizzle with brown butter.