In a large bowl or resealable container, whisk to combine the oregano, olive oil, vinegar, salt, and red pepper.
If using chicken breasts, cut each in half through their equators to create thin cutlets. Add to the marinade, mixing well to coat evenly, then cover and refrigerate until ready to cook, at least 10 minutes and up to 24 hours.
Heat a large skillet over high; a splash of water in the hot pan should result in small beads that evaporate quickly. Add 2 to 3 chicken cutlets at a time and cook until lightly browned, 1 to 3minutes per side. Transfer to a platter to rest.
Over medium-high heat, add the broccoli to the pan, season with salt, then stir-fry until charred at the edges, about 4 minutes. Add the butter and 2 tablespoons of water, and cover to finish cooking in the steam until your preferred doneness, 2 to 5 minutes.