Bring 8 quarts of water to a boil and add 3 tablespoons salt.

In a 14 to 16-inch frying pan, heat the oil over medium heat until smoking. Add the onion and cook 7 to 9 minutes, until soft and light golden brown. Remove from heat. Drop the perciatelli into the boiling water and gently stir to separate. Cook until just short of al dente, following package instructions, and drain in a colander, reserving 1 cup of the cooking water before draining. Pour the perciatelli into the frying pan with the onion and 1/2 cup of the reserved pasta cooking water and place over medium heat, adding more pasta water if necessary.

Beat the eggs in a small bowl and season with salt and pepper. Add eggs and cacio cheese to pan with the pasta and gently mix thoroughly. Watch carefully, as the eggs will start to solidify after about 1 minute. They should start to solidify, but not scramble. When they start, remove from heat and add the Parmigiano. Pour into a warmed bowl, top with pepper and serve immediately.