In a mixing bowl, whisk together all dressing ingredients until fully incorporated and smooth. (You can also blend everything in a blender until smooth).
Preheat a grill or grill pan over high heat.
Brush flat sides of avocado with oil, a squeeze of lemon juice and season generously with salt and pepper.
Place oiled sides of avocado down on grill or grill pan and sear for 3 to 4 minutes.
Remove avocados from heat and transfer to a plate.
To assemble: Arrange lettuce leaves and avocado halves onto a large platter and drizzle with dressing. Season with salt and pepper and finish with lots of shaved Parmesan. Serve with side of crostini or grilled bread.