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Pat both sides of the steak dry and season with salt and pepper.
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If using a skillet on a stovetop (preferably a cast iron skillet with vents going and windows open!), place over high heat with a mixture of oil and butter or use ghee (clarified butter). OR, if using a gas grill, turn to high to preheat.
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Put all the ingredients for the salsa verde in a food processor or blender and pulse until well combined, scraping down the sides a few times.
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Place steak in skillet and cook for about ten minutes, flipping every 30 seconds. OR place steak on the grill, cooking for about 5 minutes on each side. Steak should reach 130-135 degrees for medium rare.
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When finished, rest the steak for about 10 minutes. Meanwhile, toss the arugula with fresh lemon juice, then with a drizzle of olive oil. Season with salt and pepper. Divide evenly between plates.
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Slice the steak and place on top of the arugula. Top the steak with the salsa verde. Use a vegetable peeler to shave parmesan on top.