Bring water to boil in a large pot. Add salt, then pasta and cook until the pasta is al dente. Place a strainer or colander over a bowl or large measuring cup. Cook until the pasta is very close to ready. Drain the pasta, reserving about 1 cup cooking water. Meanwhile, add the 1/2 cup olive oil, black pepper, and 1/2 cup of pasta water scooped from the pot to a very large skillet—at least 10 inches wide, but wider is better. When the pasta is about one minute from being ready, put the skillet over a burner set to high heat. Add the drained pasta to the skillet. Reduce the heat to medium-high and cook for a couple minutes, tossing the pasta well using tongs. The sauce should stain the pasta so that it looks slippery and slick. If it starts looking dry, add a small amount of cooking water. Remove the skillet from the heat and let sit for about a minute. (This helps the cheese melt instead of clinging.) Add 1 cup Pecorino and all the Parmigiano and stir well, until the cheese is melted and evenly distributed. Drizzle with the finishing oil and toss well. Divide among four hot plates or bowls and garnish each plate evenly with the remaining 1 cup Pecorino. Serve.