Preheat a grill pan over high heat. Grill the onion rings until charred, about 2 minutes per side. If the steak is too large to fit in the pan, cut it in half. Combine 3 tablespoons olive oil, 1 teaspoon salt and 1/2 teaspoon pepper in a bowl, then brush over the meat. Grill the steak until slightly charred, 3 to 4 minutes per side for medium-rare. Transfer to a cutting board and let rest at least 5 minutes.
Meanwhile, heat the remaining 3 tablespoons olive oil in a large skillet over medium heat. Add the turnips and cook until fork-tender, about 8 minutes. Add the black-eyed peas and tomatoes and cook until warmed through, about 3 minutes. Stir in the onion rings and vinegar and remove from the heat.
Cut the steak into 3-inch-long sections along the grain, then thinly slice the meat against the grain. Stir the basil and chives into the black-eyed peas and serve with the steak.