Roast the unpeeled garlic on one side of a large skillet over medium heat, turning occasionally, until soft and blackened in spots, about 15 minutes; cool and peel. While the garlic is roasting, toast the chile pieces a few at a time on another side of the skillet, pressing them down firmly with a spatula for a few seconds, until they release a toasty aroma, then flipping them, and pressing down the other side. Scoop the chiles into a small bowl, cover with hot water and soak 30 minutes. Drain, reserving 1/3 cup of the soaking water.
In a food processor blender, combine the chiles, 1/3 cup of soaking liquid, garlic, oregano, pepper, cumin, honey, vinegar and 1 teaspoon salt. Blend to a smooth pure, scraping down and stirring frequently. Press through a medium-mesh strainer into a small bowl.
Lay the meat into a baking dish. Smear both sides of the racks with about 1/3 of the marinade. Cover and refrigerate for several hours, if time permits.
Stir enough water into the remaining marinade to give it drizzleable consistency—it usually takes at least 1/4 cup. Cover and set aside at room temperature (or refrigerate if not using within a couple of hours).
Light a gas grill, setting the temperature at medium-high on one side, medium-low on the other; or light a charcoal fire, letting the coals burn until they’re covered with white ash but still very hot, then bank them to one side. Using the side of a spoon, scrape the marinade off the meat, reserving it for basting later. Brush or spray the lamb racks with oil and lay them, meaty “cap”-side down, onto the hottest part of the grill, when they are well seared (“marked” with the grill grate pattern, as we say in the restaurant kitchen), about 2 minutes, flip them over and sear on the boney underside. Baste the top with the reserved marinade, then move them to the cooler part of the grill and let them cook until as done as you like, usually about 5 minutes for medium-rare.
It’s best to let the meat rest for a few minutes on a very cool part of the grill or in a very low oven, to reabsorb the meat’s tasty juices. Then slice the chops between the bones. Drizzle a little of the salsa on each of 4 warm dinner plates and sprinkle with the onion. Arrange the meat over the top and sprinkle with the cheese and parsley leaves. Serve right away.