1. Preheat the oven to 450°. In a medium saucepan, combine the vinegar, water, salt and sugar and bring to a rolling boil. Drop the potato slices into the vinegar mixture, cook until tender, about 5 to 10 minutes, and remove from the heat. Let the potatoes stand in the liquid while you cook the lamb.

2. Season the lamb generously with salt and pepper. Place the racks, bone-side down, on a rimmed baking sheet and roast until the exterior has browned but the center remains very rare, about 12 minutes. Remove the lamb from the oven, brush it with the vegetable oil and dredge in the cumin and cayenne mixture to coat. Roast until the lamb registers 135° on an instant-read thermometer. Let the lamb rest for 5 minutes before serving.

3. Drain the potatoes and discard the cooking liquid. Divide the potatoes among 4 plates. Slice each lamb rack in half, place the pieces on top of the potatoes and serve immediately.