Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes.
While the pasta is cooking: Heat a small, heavy-bottomed skillet over medium-high heat. Add the peppercorns and toast for about 20 to 30 seconds. Add the oil and cook for 1 minute. Remove the pan from the heat.
Drain the pasta reserving about 1/2 cup of the pasta water. Transfer the spaghetti to a large serving bowl. Add 1 cup of the pecorino and toss to combine, gradually adding reserved pasta water, if needed, to loosen up the pasta. Add the oil mixture, prosciutto, 2 tablespoons of parsley, and 2 tablespoons of basil. Toss well to combine all ingredients.
Garnish with the remaining pecorino, parsley, and basil.