For the horseradish sauce

1. Combine all sauce ingredients and stir well. Cover and refrigerate until ready to use.

For the steak sandwiches

1. Preheat grill over medium-high heat.
2. Season steak with steak seasoning and lightly brush with 1 tablespoon of olive oil. Let sit at room temperature for 20 minutes.
3. Toss onions with 2 tablespoons olive oil, salt, and pepper. Place in a grilling pan or foil packet.
4. Grill onions until softened and caramelized, about 25 minutes.
5. Place steak on grill and cook for 4-5 minutes per side, until golden brown and grill marked.
6. For medium-rare, cook until 140°F (60°C) internal temperature (another 3-4 minutes). For medium, cook until 150-155°F (66-68°C) (another 5-6 minutes). For medium-well, cook until 160-165°F (71-74°C) (another 6-7 minutes). Use a meat thermometer to ensure desired doneness.
7. Let steak rest for 3 minutes before slicing.
8. Butter the buns and grill for 1-2 minutes, butter side down, until lightly toasted. Watch closely to prevent burning.
9. Cut the rested steak thinly.
10. Spread approximately 2 tablespoons of horseradish sauce on the bottom bun.
11. Top with steak and onions.
12. Top with arugula and the top bun.
13. Serve.