Step 1Use paper towels to pat the steak completely dry. Generously season both sides of the steak with salt and pepper, then rest the steak at room temperature for at least 15 minutes.
Step 2Meanwhile, in a small pot over low heat, combine the cherries, sherry vinegar, and a pinch of salt. Warm for a few minutes, then remove from heat and set aside to soak while you cook the steak.
Step 3Heat a large cast iron skillet over medium-high heat until it’s so hot it’s nearly smoking, 3 to 4 minutes. Swirl in 1 tablespoon of the olive oil. Place the steak in the skillet and cook until browned on the first side, about 4 minutes. Flip and cook on the second side to your preferred doneness, 3 to 5 minutes for medium-rare, depending on thickness. (If using a thermometer to check doneness, 130 degrees is medium-rare.) Use tongs to hold the steak on its side and crisp any strips of fat for another minute or so. Transfer the steak to a plate and let rest for 10 minutes.
Step 4Scatter the arugula across a serving plate. Using a fork, stir the remaining 3 tablespoons of olive oil into the sherried cherries. Thinly slice the steak against the grain and place the slices on the arugula. Drizzle any juices from the cut steak over the top, then spoon the cherries and their sauce over everything.