Prepare your grill to cook at medium heat. Cut vegetables into bite size chunks and mix with Arbequina olive oil and a sprinkle of salt. Place the vegetables on the grill. After the first ones laid down have cooked for about two minutes rotate them about 45 degrees without turning them. Cook for another minute or two and then flip or turn them to another un-grilled surface. Cook for another 2 minutes and rotate them again. Continue grilling the vegetables until they are cooked but still somewhat firm. Remove all of the vegetables from the grill and set aside to cool.
In a food processor, add in the basil, nuts and cheese and pulse a few times. Add the oil in a slow stream through the feeder tube until it is all incorporated. Season to taste with salt.
When the vegetables have cooled, cut them all into bite sized pieces. Combine the vegetables in a large bowl. Add the pesto to the vegetables and stir to combine. Season with additional salt, if necessary. Place the vegetables into a serving bowl and sprinkle with parmesan cheese.