Light a gas or charcoal grill to high heat. When hot, add corn and cook on all sides until some kernels have started to blacken, 5 to 8 minutes. Remove corn from the grill to cool. Cut kernels off the cob and place in a medium bowl.
Halve and pit avocado, then scoop flesh from the peel and dice. Add diced avocado, salt, pepper, and 1 tablespoon of dressing to the bowl with the corn, stirring gently to combine. Set aside.
Drizzle the cut side of both romaine halves with about 2 teaspoons of olive oil. Season with salt and pepper to taste. Grill romaine, cut side down, for 1 to 2 minutes. Flip, and cook for an additional 1 to 2 minutes. Remove from heat, cut off root ends, and place cut side up on a serving platter or bowl.
Drizzle romaine with about ¼ cup of the dressing and top with corn-avocado mixture and shaved SarVecchio. Drizzle with additional dressing and serve (you’ll have leftover dressing).