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Season the steaks generously with salt and pepper and let sit at room temperature 1 hour.
Meanwhile, make the sauce: in a small bowl, combine the sour cream, horseradish, half of the lemon juice, and one-quarter of the chives. Season with a generous pinch of pepper.
Heat a dry medium skillet, preferably cast-iron, over medium-high, then add 1 teaspoon of the oil. As soon as the oil is smoking, cook the steaks, turning every 1-1/2 to 2 minutes, until a deep brown crust forms, 8 to 10 minutes. Reduce the heat to medium and add the butter. Tilt the pan toward you so the butter pools on one side, then use a large spoon to continually baste the steaks with the butter. Continue until the butter is no longer bubbling and it smells nutty and is beginning to brown, about 1 minute. Transfer the meat to a cutting board and let it rest 10 minutes. (A medium-rare steak should reach an internal temperature of 120°F to 125°F). Add any collected juices from the meat to the brown butter, then keep the butter warm.
In a medium bowl, combine the zucchini, fennel, and Pecorino with the remaining lemon juice, the remaining olive oil and chives, and the salt and pepper. Toss to combine.
Transfer the steaks to dinner plates, spoon the browned butter over the steaks, and sprinkle the dish with salt and pepper. Add the zucchini and the horseradish cream sauce on the side and enjoy.
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