Preheat the oven to 375 degrees F.
Trim any excess fat on the lamb racks and season, to taste, with salt and pepper. Heat 2 tablespoons of olive oil in a large ovenproof skillet over medium-high heat and put the racks in the hot pan, fat side up. Sear until browned, then flip and sear on the fat side until golden brown. Stand the racks up in the pan, criss-crossing the bones with the fatty side out. Put the pan in the oven to roast until the internal temperature reaches 130 degrees F on an instant-read thermometer, about 15 to 20 minutes. Remove the pan from the oven and let the lamb rest on a cutting board for at least 5 minutes. Slice into single or double chops (2 rib bones).
To make the sauce, finely chop the pineapple, reserving any juices that are released (it should resemble canned crushed pineapple). Heat the remaining 1 tablespoon of olive oil in a saucepan over medium heat. Add the garlic and cook until softened, 2 to 3 minutes. Add the pineapple and its juice, the mint bundle and water to the saucepan and simmer over low heat until the pineapple is broken down and the liquid has thickened, about 40 to 45 minutes. Remove the mint bundle and discard; stir the chopped mint into the sauce and transfer to a serving bowl.
Serve the lamb chops on a platter topped with the pineapple mint sauce spooned over the top.