Pound the steak to flatten and set aside. Sauté the onion, mushrooms, garlic and greens in olive oil until softened and most of the liquid is evaporated. Sprinkle the cheese curds along the steak, then top with the greens mixture using a slotted spoon. Carefully roll the steak back up. To portion the steak, tie twine around the middle, then tie twine halfway between each end. Cut along the middle to create two halves. Then cut along each of the other strings, ending up with 4 pieces.
Lay out a bit of foil over the grill, so you don't lose any of the cheesy goodness. Grill on medium heat, turning frequently to cook each side. Grill to 160°. Use a meat thermometer to determine doneness. Once you get the pinwheels off the grill, give them a few minutes before you serve, so the steak will retain its juices.