Toss sliced carrots with 2 tsp. salt and put in a colander. Put a plate on top of carrots, weigh down with a 1-lb. can, and set in sink to drain for 1 hour.
Meanwhile, chop garlic, sprinkle with a pinch of salt, and mash to a paste on a cutting board with the flat side of a chef's knife. In a small frying pan, toast cumin seeds over medium heat until fragrant, 1 minute, and grind finely with a mortar and pestle (or put cooled seeds in a resealable plastic bag and crush with a meat mallet or the underside of a small heavy frying pan). In a small bowl, whisk together garlic, cumin, orange juice, sherry vinegar, honey, pepper, remaining 1/4 tsp. salt, and minced cilantro; whisk in olive oil.
Rinse carrots and turn onto a clean dish towel. Gently dry with another towel. In a large bowl, toss carrots with half of the vinaigrette and the shaved cheese. Divide lettuce leaves among plates and drizzle lightly with remaining vinaigrette. Layer carrots on top and crown with cilantro leaves.
Note: Nutritional analysis is per serving.