Bring a large pot of water to a boil. Preheat the broiler and position a rack 5 to 6 inches from the heat source. Arrange the lamb chops in an ovenproof skillet or a rimmed baking sheet and drizzle some EVOO over them. Season the chops liberally with salt and pepper on both sides and reserve.
Salt the water and cook the spaghetti until al dente. While the pasta cooks, in a large bowl combine 1/3 cup of EVOO with the balsamic vinegar, soy sauce, garlic, scallions and mint, and toss with the tomatoes.
Place the lamb chops under the broiler and cook for 4 minutes on each side for rare or 5 minutes on each side for medium-rare. Let the meat rest for a few minutes before serving.
Drain the pasta well, add to the tomatoes and toss. Scatter in the cheese and grind lots of pepper over the pasta. Toss again and add salt to taste. Serve a pile of the spaghetti salad alongside a couple of chops.