Heat grill to high.

Wash eggplants and then poke them thoroughly. If you don't poke the skin the steam will build up in the eggplant while grilling it and it will split and make a mess.

Grill the eggplant, turning every 5 minutes or so, at least 25 minutes for large eggplants. Check smaller ones for doneness starting at 15 minutes. You want the eggplants to be charred all over their surface and so soft that you are worried about being able to pick them up and put them in the bowl. (If you are really worried use a nice, wide spatula to scoop them up.) If you are not sure they are soft enough, give them another 5 minutes because there is nothing worse than undercooked baba ghanoush.

Once they are finished grilling, transfer the eggplants to a colander set over a bowl. Let them drain and cool for 20-30 minutes.

When the eggplants are cool enough to work with, start splitting them open and scraping out the flesh from the skins. I do this in the colander to allow more of the juices to drain off.

Put peeled eggplant, garlic, lemon juice, salt, mayonnaise, and olive oil in the food processor. Pulse ingredients until the dip is as coarse or as smooth as you like. Start with a coarser, choppier texture, about 8 or 9 pulses, and try it before processing it more.

Adjust salt to taste. Transfer dip to a serving bowl, drizzle some olive oil on top, and sprinkle the parsley over it. (The dip can be made one day ahead - cover with plastic wrap and put in the fridge. Allow to warm up to room temperature before serving.)