In a small bowl using a fork, blend 4 tablespoons of the butter, the cheese, parsley, ¼ teaspoon salt, and ¼ teaspoon pepper. Set aside until ready to serve.
Lightly brush the steaks on both sides with oil and season with 2 teaspoons salt and ¾ teaspoon pepper. Let stand at room temperature for 15 to 30 minutes before grilling.
In a large skillet add 1 tablespoon oil and the remaining 1 tablespoon butter. Cook over medium heat until the butter is melted. Add the onions and 1 teaspoon salt and mix well. Cook until the onions soften and turn golden brown, stirring occasionally, about 20 minutes. Reduce the heat to low, add the vinegar and ¼ teaspoon pepper, and continue to cook until the onions are very tender, 20 to 30 minutes more, stirring frequently.
Prepare the grill for direct cooking over high heat (450° to 550°F).
Grill the steaks over direct high heat, with the lid closed, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once or twice. Remove from the grill and let rest for 3 to 5 minutes. Serve the steaks warm with a spoonful of butter on top and the onions on the side.