Whisk together ⅓ cup olive oil, next 3 ingredients, and remaining ½ teaspoon salt and set aside.
Brush eggplant slices evenly with remaining 2 tablespoons olive oil grill in a Weber charcoal or gas barbecue, covered with grill lid, over high heat (400°F to 500°F) 2 minutes on each side.
Place eggplant in a large shallow dish and drizzle with half of olive oil mixture.
Grill avocado slices, without grill lid, 2 minutes on each side.