Melt the sugar with 1 tablespoon water in a medium saucepan and cook over medium heat, swirling the pan occasionally, until a golden caramel is formed, about 10 minutes. Add the lime juice, bring to a boil and allow the sugar to melt into the sauce until reduced by half, about 5 minutes. Add the broth and boil until reduced to 2/3 cup, about 3 minutes. Season with 1/2 teaspoon salt.
Preheat the oven to 220°C. Steam the carrots in a covered rack set over boiling water until tender, about 15 minutes. Stir together 3 tablespoons of the key lime gastrique and 1/4 teaspoon each salt and pepper. Toss the carrots with 1 tablespoon of the butter, the seasoned gastrique and the parsley.
Stir together the coriander with 1 teaspoon salt and 1/2 teaspoon pepper. Rub the spice mixture evenly over the lamb.
Melt the remaining 1 tablespoon butter in a large, heavy, ovenproof skillet over medium heat until hot. Sear the lamb on two sides until browned, about 2 minutes each side. Transfer the skillet to the oven and roast until the lamb is medium, 20 to 25 minutes.
Let the lamb stand 10 minutes before serving with the carrots and remaining gastrique.
Cook's Note: For New Zealand lamb chops, serve 4 chops per person. For American lamb chops, usually 2 chops per person is a nice portion.