If you're tight on time, coarsely grate carrots in a food processor instead of peeling into long strips.
Rub lamb with garlic, and season with 1 teaspoon salt and 1/2 teaspoon pepper. Grill on medium heat 4–5 minutes per side (medium-rare) or to desired doneness. Let rest 5 minutes, then cut into bite-size pieces, discarding bones.
Whisk together yogurt, lemon juice, cinnamon, cayenne, and remaining 1/2 teaspoon salt in a large bowl.
Use a vegetable peeler to cut carrots into long, thin strips. Add carrots and the remaining ingredients to dressing, and toss. Serve with lamb.