In a medium bowl, combine the dried basil, red pepper flakes and lemon zest. Add the chicken and season with salt and pepper. Marinate, refrigerated, for 45 minutes.
Meanwhile, preheat the oven to 425 degrees F. Once hot, put a baking sheet lined with foil in the oven for 20 minutes to heat up.
Add the onion, carrots, bell pepper and broccoli to the bowl with the chicken, drizzle with a little olive oil, sprinkle with some salt and pepper and toss. Put the chicken and vegetables flat on the hot baking sheet and sprinkle with more salt and pepper. Put the baking sheet back in the oven and roast, flipping the chicken halfway through to promote even browning, until the chicken is fully cooked, about 20 minutes.
Transfer the chicken and vegetables to a platter, sprinkle on the Pecorino Romano and pickled cherry peppers and drizzle with olive oil.