Make the chimichurri: Combine all of the herbs in a medium bowl. Add the oil, garlic and chile flakes and stir to combine. Let sit for at least 30 minutes to allow flavors to meld. Stir in the vinegar just before serving to keep the color bright green. Season with salt and pepper, then taste and add honey, if needed.
Meanwhile, prepare the steak: Let the steak sit at room temperature for 30 minutes before cooking. This will help it cook quickly and more evenly.
Preheat the broiler. Heat a large ovenproof skillet, preferably cast-iron, over medium-high heat, then heat the oil until smoking. Season the steak very generously with salt and pepper and cook just on one side, undisturbed, until a deep brown crust forms on the bottom, about 4 minutes. Transfer the steak, browned-side up, to a cutting board.
Cut the steak off the bone in 2 pieces--the strip steak on one side and filet mignon on the other. Slice both pieces 1 inch thick straight down, perpendicular to the bone. Return the bone to the skillet and reassemble the sliced steak around it, browned-side up, to resemble the whole steak. Top with the butter and broil until the butter is melted and golden brown and the steak is medium-rare, 4 to 6 minutes, spooning some of the butter and pan juices over the steak as it cooks. Scatter the cheese over the top and broil for 1 more minute. Serve the steak with buttery pan juices spooned over and the chimichurri on the side.