Cut onions in half. Slice into strips about 1/4″ wide.
Place the onion in a large skillet and cover with 1/4” of water. Bring to a boil and cook until the water has almost evaporated.
Reduce the heat to medium low. Add 1 tablespoon of the olive oil and 1 tablespoon of butter to the pan. Saute the onions on a low heat for about 10 minutes, until they are reduced and starting to caramelize.
To make the aioli, mix together mayonnaise, garlic, and lemon juice in a small bowl.
To make the chimichurri, add parsley or cilantro, 1/4 cup olive oil, red wine vinegar, a red chili, oregano, 1/4 teaspoon of salt, and pepper to a chopper or food processor. Blend until nearly smooth.
Preheat a grill pan to high heat.
Brush the steak on both sides with the remaining olive oil. Season with remaining salt and ground black pepper.
Place steaks on the grill. Sear for 3-5 minutes per side, depending on thickness. Test with an instant read meat thermometer and remove the steak when it reaches 125°-130° F (45°-50 °C)for medium rare (or your preferred doneness). Rest the steak for 10 minutes before slicing against the grain.
Meanwhile, butter the rolls and set them on the grill for 2-3 minutes. Transfer to a plate.
Spoon a generous amount of aioli on each side of the roll. On the bottom half of the roll, spread the steak out along the roll and top with caramelized onion and a slice of Swiss cheese. Set the bottom half on a baking sheet and put it in the oven on broil for 2-3 minutes, just to melt the cheese on top.
Remove from the oven. Top with arugula, chimichurri, and the top bun. Enjoy.