Bring a large pot of water to a boil. Salt it generously, then stir in the pasta and cook according to the package instructions. During the last 2 minutes of cook time, add the broccoli. Reserve about 1 cup of the pasta water, then drain and transfer back to the pot.
Add the chicken to a bowl. Season all over with pepper and salt, drizzle with a touch of olive oil, then use your hands and toss to coat.
In a large skillet over medium-high heat, warm 1 tablespoon oil. Once hot, add the chicken and cook until nicely browned, about 5 minutes. Turn the heat down to medium-low, flip the chicken and cook for 5-10 more minutes or until the chicken is cooked through (covering with a lid can speed this along). Transfer to a cutting board and chop it into bite-sized pieces.
To the pot with the pasta/broccoli, add the chicken, pesto, and about ¼ cup of pasta water. Stir until everything is combined and coated in the sauce. You can thin it out with more pasta water (or just hot water), if needed. Stir in the remaining 2 cups of arugula.
Serve right away topped with plenty of parmesan and black pepper.