Heat oven to 400° F.
Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the lamb with 3/4 teaspoon of the salt and 3/4 teaspoon of the pepper. Add 4 of the chops to the skillet and cook until browned, about 2 minutes per side. Transfer to a baking sheet. Repeat with the remaining chops. Roast to the desired doneness, 4 to 6 minutes for medium-rare.
Meanwhile, in a medium bowl, combine the tomatoes, garlic, fennel, lemon juice, oregano, and the remaining oil, salt, and pepper. Divide the arugula and lamb chops among individual plates and spoon the tomato mixture over the greens.
Tip: The bright, refreshing flavors of the tomato, fennel, and arugula salad also go well with seared steak or salmon.