Heat a grill to medium (350° to 450°). In a small bowl, combine 1 tbsp. oil with the cayenne. Rub romaine with oil mixture. Grill, turning once, until slightly softened and streaked brown, 5 to 6 minutes total. Arrange romaine cut side up on a platter.
Whirl remaining 2 tsp. oil, the yogurt, lime juice, avocado, garlic, cilantro, salt, and 2 1/2 tbsp. water in a food processor until smooth. Spoon dressing over romaine and sprinkle with cheese and green onions.
*Find in grocery stores and Latino markets.