Preheat oven to 375 degrees F. Bring water for pasta to a boil.
Season chicken on both sides with salt and pepper. Heat a large oven-safe skillet on medium heat and add 2 tablespoons of olive oil. Once hot, add chicken and cook until golden brown on both sides, about 5 minutes each.
While the chicken is cooking, add pistachios, basil, parsley, garlic and cheese to a food processor. Pulse and blend until the nuts are coarsely chopped, then with the processor on, stream in 1/3 cup of olive oil. Add a little more if necessary, depending on how “spreadable” you want your pesto to be. Finish by adding in salt and blending once more. Spread a thin layer of pesto on top of each chicken breast in the skillet, then place in the oven and bake for 15 minutes.
While chicken is baking, boil pasta and prepare according to directions. Once finished, drain pasta and toss with remaining 2 tablespoons of olive oil, about 1/4 cup of pesto, the red pepper flakes and grated pecorino. You just want a very light coating of pesto on the noodles. Once chicken is done, remove and spread a little additional pesto on top. Serve together!