Pre-heat the oven to 400 degrees to roast the broccoli. Get a large pot of water on the stove to boil for the pasta. Transfer the broccoli florets into a baking dish and give it a generous pinch of salt. Drizzle the broccoli liberally with olive oil and toss everything together. Roast it in the oven for 10 minutes, until tender.

In the meantime get a large pot on the stove and heat a dash of olive oil in it over medium high heat. Season the diced chicken with salt, then get them into the hot pan. Cook the chicken until it is completely cooked through and slightly brown for about 5 minutes, then remove it all from the pan and set it aside. In the same pan, combine the cup of olive oil, lemon olive oil (or juice), garlic, parsley, crushed red pepper, and salt to taste. Let the aglio olio sauce come to a low boil then quickly turn down the heat to low to let the flavors gently infuse for 5 minutes. Take the broccoli out of the oven at this point and add it to the pan, along with the chicken chunks. Let all of the flavors meld together on low for another 15 minutes.

While the sauce is cooking, salt the boiling pasta water until it tastes like the sea. Pour the pasta in the boiling water and cook it until it is just tender for about 8 minutes. Reserve 1/4 cup of the pasta cooking water and pour it into the pan of sauce. Drain the pasta and add it to the pan of aglio olio sauce and toss everything thoroughly to combine. Take the pan off of the heat and serve the pasta immediately!