In a dish large enough to hold the lamb chops in a single layer, combine half the lemon juice, 1/3 cup olive oil, the rosemary, garlic, 1 teaspoon of the salt and pepper. Add the lamb, turn to coat with the marinade and set aside for about 15 minutes at room temperature or up to 2 hours in the refrigerator.
Preheat a large skillet over medium-high heat, for about 1 minute. Pat the chops dry and season on 1 side with salt and pepper. Add enough oil to lightly coat the surface of the pan. Working in batches if needed, add the chops seasoned side down to the pan. Cook until crisp and brown, about 2 to 3 minutes. Season the top side with salt and pepper, turn, and continue cooking until just firm and an instant-read thermometer registers 130 to 135 degrees F, about 1 to 2 minutes. Transfer to a platter and tent with foil to keep warm. Allow the chops to rest for 5 to 10 minutes.
Pour off any fat left in the pan. Return the skillet to medium heat. Add the chicken stock, and scrape up the brown bits on the bottom of the pan with a wooden spoon. Add half the remaining lemon juice and simmer until the mixture is reduced to a glaze, about 4 minutes. Swirl in 2 tablespoons of butter. Add the basil and olives and season. Return the lamb chops to the sauce and toss and turn to coat thoroughly with sauce. Arrange chops on a platter, pour sauce over shops and garnish with basil leaves.
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