CARAMELIZE THE ONIONS: Slice the onions and cook them in butter over medium heat until they are caramelized, about 40 minutes. (Use a shortcut method if you can't wait.)
PREPARE THE SAUCE: In a blender, combine the caramelized onions, olive oil, egg yolks, grated parmesan cheese, water, salt, and pepper. Blend until smooth. Add more water to thin it out a little if needed. I would also recommend adding the water in parts throughout the blending process, so you don't get a runny sauce.
COMBINE AND HEAT: Toss the cooked pasta with the sauce in a large pan. Warm over low heat for about 1 minute, then turn off the heat.
SERVE: Garnish with fresh parsley if desired.
