Bring water to a boil for pasta, salt water and cook to al dente.
Toast walnuts in a dry skillet over medium heat until golden and fragrant, a few minutes then cool. Add the nuts to a food processor with rosemary, 1 clove garlic, salt and pepper and turn processor on. Stream 1/3 cup oil into food processor to form pesto then scrape into a bowl and stir in about 3/4 cup grated cheese.
Rub chops with cut clove of garlic then season with salt and pepper. Heat 1 tablespoon oil in a large skillet over medium-high heat. Add chops and cook 4 minutes each side for medium rare doneness.
While lamb cooks, drain pasta and return pot to heat. Add 1 tablespoon oil and butter in pasta pot and melt over medium heat. Place the pasta back in the pot and crisp 4 to 5 minutes, toasting the pasta. Season the pasta with salt and pepper and a little nutmeg.
Remove lamb from skillet and cover with foil to let meat settle. Heat remaining 1 tablespoon oil over medium-high to high heat. Add onions and greens and sear them. Season the greens with salt and pepper and wilt down, 3 to 4 minutes. Add vinegar and turn greens to coat in vinegar.
Serve a chop alongside greens. Top greens with toasted pasta and drizzle some pesto over everything.
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