Preheat oven to 450°F. Combine apple juice and cornstarch in a small bowl; set aside.
Sprinkle lamb chops with salt and pepper. Heat oil in a large ovenproof skillet over medium-high heat. Add the chops and cook until browned on one side, about 2 minutes. Turn them over and transfer the pan to the oven. Roast until an instant-read thermometer inserted horizontally into a chop registers 140°F for medium-rare, 6 to 10 minutes, depending on the thickness. Transfer the chops to a plate and tent with foil.
Place the skillet over medium-high heat (take care, the handle will still be hot). Add shallot and cook, stirring constantly, until browned and softened, about 1 minute. Add broth, vinegar and jelly and bring to a boil, whisking to dissolve the jelly. Cook, whisking constantly, until the liquid has reduced by half, 2 to 3 minutes. Stir the cornstarch mixture; add to the pan and bring to a simmer, stirring constantly, until the sauce is thickened, about 30 seconds. Remove from the heat and stir in half the mint along with any accumulated juices from the chops. Serve the chops topped with the sauce and the remaining mint.