In a mixing bowl, combine the room temperature butter with the grated Parmesan cheese. Mix until well blended. Set aside.
Pat the rib-eye steak dry with paper towels. Season generously with salt and pepper on both sides to enhance the flavor.
Heat the olive oil in a skillet or grill pan over medium-high heat. Once hot, add the steak and cook for about 4-5 minutes on each side for medium-rare, or until it reaches your desired doneness. Use a meat thermometer to ensure accuracy.
While the steak is cooking, prepare the balsamic glaze. In a small saucepan, combine the balsamic vinegar, chopped shallots, and dark brown sugar. Bring to a simmer over medium heat and let it reduce for about 8-10 minutes, or until it thickens slightly.
Once cooked to your liking, remove the steak from the heat and let it rest on a cutting board for about 5 minutes. This allows the juices to redistribute, making the steak even more tender and juicy.
Arrange the arugula or mixed greens on a platter. Slice the steak against the grain and place it on top of the greens. Drizzle the balsamic glaze over the steak, and add dollops of the Parmesan butter. Finish with shavings of Parmesan cheese and serve with lemon wedges on the side.