In medium high-sided skillet, heat oil and pepper on medium-low until pepper starts to sizzle, about 2 minutes; remove from heat and stir in lemon zest.
Place pasta in large skillet and add enough water to cover completely (about 4 cups). Bring to a boil. Add 4 tablespoons lemon juice and ½ teaspoon salt and cook, separating pasta and stirring with tongs, until just tender, 7 to 10 minutes.
Stir ⅓ cup pasta cooking water into pepper oil; then, using tongs, transfer pasta to skillet and toss, adding both cheeses in 6 increments and stirring vigorously after each addition. Stir in 1 to 2 more tablespoon pasta water as needed until cheese melts into a creamy sauce and fully coats pasta. Drizzle with remaining tablespoon lemon juice and season with ¼ teaspoon each salt and cracked pepper. Serve immediately.
