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Make the horseradish sauce by combining all sauce ingredients and stirring well to combine. Cover and let sit in refrigerator until ready to use.
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Heat grill to medium-high heat.
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Season steak with steak seasoning. Lightly brush with 1 tablespoon of olive oil. Let sit at room temperature for 20 minutes.
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Meanwhile, toss onions with additional 2 tablespoons olive oil and season them with salt and pepper.
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Place onions in a grilling pan or wrap in foil. Place on grill and cook until onions are softened and beginning to caramelize, about 25 minutes.
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Once steak has sat at room temperature for 20 minutes, place on grill.
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Cook for 4-5 minutes, until steak is golden and has grill marks, and then flip. For medium-rare, cook until internal temperature reaches 140 F (another 3-4 minutes); for medium, cook until 150-155 F (another 5-6 minutes); and for medium-well cook until 160-165 F (another 6-7 minutes). This always depends on the thickness of your steak. To be sure, use a meat thermometer.
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Let your steak rest for 3 minutes before slicing.
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While steak is resting, butter the buns.
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Place bun halves on grill, butter side down, and lightly grill for 1-2 minutes. Keep a close eye on them, as they will burn easily.
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Once steak has rested, cut it as thinly as possible.
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Place approximately 2 tablespoons of spread on the bottom half of each bun.
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Top with steak and onions.
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Top with arugula and finally the top half of the bun. Serve and enjoy!