In a 12 to 14-inch saute pan, heat olive oil until smoking. Add onions and potatoes and cook until softened and lightly browned, about 8 to 10 minutes. Add vinegar and sugar and bring to boil. Heat and reduce until almost dry, about 15 minutes. Allow to cool and stir in mint leaves.
Roll out pasta to thinnest setting on pasta roller. Cut dough into 3-inch disks and place 1 tablespoon onion mixture in center of half the disks. Use the other half to cover and pinch sides together. Continue until all filling is used up.
Bring 6 quarts water to boil and add 2 tablespoons salt. Add culingionis and cook until tender, about 3 to 4 minutes. Meanwhile, melt butter in a 12 to 14-inch saute pan. Drain pasta carefully and place in pan with butter. Add mint leaves and half the cheese and toss. Serve immediately.
Place both types of flour in a large mixing bowl and stir to mix well. Make a well in the center of the flour mixture and add water a little at a time, stirring with your hands until a dough is formed. You may need more or less water, depending on the humidity in your kitchen.
Place the dough on a floured work surface and knead it like bread for 8 to 10 minutes until smooth and elastic. Cover and let stand for 10 minutes at room temperature. Yield: 4 servings
Recipe courtesy of Mario Batali .