In a large bowl, combine eggs, flour, salt, 1 tablespoon oil, and 1/2 cup water. Mix until smooth, then let rest for 10 minutes. Meanwhile, heat 1/4 cup oil in a skillet over medium heat. Saute onion slices until golden brown; set aside. Preheat oven to 300 degrees F (150 degrees C).
Bring a large pot of lightly salted water to a boil. Place 1/3 of the dough into a spaetzle maker or coarse sieve or collander with holes about 4 millimeters in diameter. Let dough drop into boiling water. Boil until spaetzle rises to the top, then transfer to a 9 inch casserole dish with a slotted spoon. Cover with 1/3 of the cheese. Repeat layers with remaining spaetzle and cheese. Spoon fried onions over top.
Bake in preheated oven for 15 minutes, or until cheese is thoroughly melted. Before serving, sprinkle with 1 or 2 tablespoons vinegar, and sprinkle with chopped parsley.