Heat the butter in a large skillet set over medium heat. Once the butter has melted, add the onions and cook, stirring occasionally for 20 minutes. Reduce the heat to medium-low, then stir in the onion soup mix. Cook the onions, stirring as needed to prevent burning, until they are dark brown and caramelized, about an additional 30 minutes. If at any point the onions begin to burn, add 2 tablespoons water to the pan and continue cooking. Set the onions aside.
Season the steaks with salt and pepper. Coat the bottom of a large skillet set over medium-high heat with olive oil. Once the skillet is hot, add the steaks and cook to your desired doneness. For medium-rare, cook for about 8 minutes, or until they reach an internal temperature of 130°F.
Transfer the steaks to a cutting board and let them rest for 10 minutes. Slice the steaks as thin as possible against the grain.
Preheat the oven to broil. Place the four bottom halves of the sandwich rolls on a baking sheet, then divide the onions among the rolls. Top the onions with the steak and Gruyère cheese. Place the sandwiches under the broiler until the cheese is melted.
In a medium bowl, whisk together 2 teaspoons olive oil and the lemon juice. Add the arugula and a pinch of salt and pepper, then toss to combine. Top the sandwiches with the arugula, then add the top halves of the rolls and serve.