Mix all the ingredients for marinade into a thick paste. Place the chicken breasts in a flat dish and coat generously with the marinade. Let it rest for 2 hours or up to 24 hours in the fridge.
Heat the olive oil in a pan on medium heat. When the oil is quite hot but not smoking, place the chicken breasts in the pan (they should be flat, with maximum surface area touching the hot pan), scatter the broccoli florets on top, cover and cook for 4-5 minutes.
After 4 mins, flip the chicken breasts, very lightly sprinkle a little salt over the broccoli, cover and cook for a further 4 mins. The chicken should be browned and cooked through at the end of 4 mins. If a knife inserted into the thickest part of the meat releases clear juices, it's done.
Serve the chicken and broccoli warm with pan juices drizzled on top.