Add pasta to a large pot of boiling salted water; return to boil. Cook for 10 to 12 minutes or until al dente. Drain; reserve 1 cup pasta water.

Pat chicken dry with paper towels. Melt 2 tablespoons butter in a small microwave-safe bowl. Stir in blackened seasoning and salt. Coat chicken pieces with butter mixture.

Melt 2 tablespoons butter in a large nonstick skillet over medium heat. Cook chicken 15 to 18 minutes or until chicken reaches 165°, turning halfway through. Transfer to a cutting board. Loosely cover and let stand for 5 minutes. Wipe skillet clean.

Melt 2 tablespoons butter in same skillet over medium heat. Add 1 teaspoon lemon zest and juice and 2 teaspoons lemon-pepper. Cook and stir for 1 minute. Add reserved pasta water; bring to a simmer over medium heat. Add pasta and remaining 2 tablespoons butter. Remove from heat. Add ½ cup Parmesan and Asiago cheeses; toss to combine.

Cut chicken pieces crosswise on an angle into ¼-inch-thick slices; add chicken and arugula to pasta mixture. Toss to combine. Divide among four serving plates. Garnish with additional lemon zest, lemon-pepper seasoning, and grated Parmesan cheese, if desired.