Bring a large pot of salted water to a boil and cook the spaghetti al dente according to package directions. Reserve 2 cups of pasta water and drain.
Add 4 tablespoons of the butter to a large, straight-sided pan and heat over medium. Once melted, add the pepper and toast for about 1 minute and 30 seconds. Add in about 1 1/4 cups of the reserved pasta water and the lemon zest, bring to a light simmer. Turn the heat down to medium-low and add the cooked spaghetti, Pecorino Romano, Parmigiano-Reggiano and the remaining 2 tablespoons of butter. Using tongs or two wooden spoons, toss the pasta and cheese together, incorporating about 1/2 additional cup more pasta water. Keep tossing the pasta until the cheese has completely melted and the sauce is silky and coating the pasta, 3 to 5 minutes. Add the lemon juice and chives and toss to combine. Serve immediately, topped with extra Pecorino Romano and black pepper.