Bring a large pot of salted water to a boil. Cook pasta al dente according to package directions. Drain; reserving ½ cup pasta water.

In a large skillet, melt 2 tablespoons of butter. Add pepper and toast until fragrant. Add pasta water, spaghetti, and remaining tablespoon of butter. Over low heat add cheese and use tongs to toss until pasta is coated and cheese has melted. Turn off heat and toss in arugula and lemon zest. Transfer to bowls and serve warm.