If you have a grill, pat yourself on the back and fire it up. Make sure the eggplants are pricked all over with a fork, then grill them until they have softened considerably and are charred all over. If you do not have a grill, consider yourself a revolutionary and char the eggplants over a gas burner. While charring, turn the eggplants occasionally to make sure all side are blackened. Heat the oven to 350*F and place the eggplants and the head of garlic on a sheet pan. Drizzle everything with olive oil and a fat pinch of salt. Finish roasting the eggplants another 20 minutes, until they have completely softened on the inside. Set aside and let cool. Continue to let the garlic roast another 15 minutes or so, until the cloves are soft.
Once the eggplants have cooled, scrape out the insides with a spoon into a bowl. I find that this step is not so much scraping the insides out as it is peeling away the skin and discarding. Remove the roasted garlic cloves from their skins and mash with the back of a knife. Add the roasted garlic to the bowl with the eggplant, as well as the tahini, fresh minced garlic, Greek yogurt, cumin, cayenne, paprika, a pinch of sea salt and the lemon juice. Transfer this to a food processor or use a hand blender and mix well, until the dip is a creamy consistency.
Set aside for five or six hours. After the dip has sat and the flavors have melded, re-season with sea salt and lime juice as needed before serving.