Slice each avocado in half, lengthwise. Remove the pit. This avocado slicer makes this super easy.
Brush each avocado with olive oil and place them face down on the grill. Close the lid.
The avocados should have nice grill lines after about ten minutes, but start checking them after five minutes to be safe. You can carefully use an oven glove and a spatula to gently lift one up to check out the bottom, then put it back in place if it needs more time.
Once nice and grilled, place the avocados on an aluminum tray and leave the grill on.
Crack an egg into a small bowl or cup and use a spoon to lift out the yolk, place it in the middle of each avocado. If you have extra room in the middle, add some additional egg white. Sprinkle with salt/pepper.
Place the aluminum tray on the grill and close the lid. Bake for about 12 minutes, until the yolk is set and cooked to your preference. Use a spatula to place them on a plate, garnish with parsley if desired, and serve!