Preheat a grill or grill pan.
Place peeled and pitted avocado halves in a bowl. Drizzle with olive oil and lemon juice (to prevent oxidation), and sprinkle with a good amount of grey salt and black pepper. To grill indoors, preheat a grill pan over medium heat. Douse a paper towel with olive oil, then swipe it across the grill pan so you have just enough olive oil so the avocados won't stick (they will supply their own fat). Place avocados on the grill, cut side down. Grill for 1 full minute.
Meanwhile, preheat a 10-inch saute pan. Add 1/2 cup of olive oil to the pan, add minced garlic and saute until it gets light brown. Add julienned basil and saute until wilted and transfer to a large glass or ceramic bowl. Take the avocados off the grill and add to the bowl. With a fork, mash the avocado/basil mixture well. Add sour cream and stir. Add Parmesan and stir. Serve immediately.