Position a rack 4 inches from the broiler element and heat the broiler to high. Line a rimmed baking sheet with foil and grease lightly with oil. Rub the eggplant all over with 2 Tbs. of the oil and sprinkle the flesh side with 1/2 tsp. of the salt and the 1/4 tsp. pepper. Arrange the eggplant, flesh side down, on the baking sheet and broil until the skin is charred and the eggplant flesh is very tender, 20 to 30 minutes.
Meanwhile, if using anchovies, mash them into a paste with the side of a chef’s knife. Roughly chop the garlic, sprinkle it with a generous pinch of kosher salt, and mash it into a paste with the side of a chef’s knife. Transfer the anchovy and garlic pastes to a food processor and add the beans, 2 Tbs. of the oil, and 1 Tbs. water. Purée until smooth.
When the eggplant is done, set it aside to cool briefly. Meanwhile, in a medium bowl, toss the pita wedges with the remaining 1 Tbs. oil and 1/4 tsp. salt. Arrange in a single layer on a baking sheet. Lower the rack so it’s 6 inches from the broiler. Broil the pita wedges until golden brown on both sides, 1 to 2 minutes per side.
Scrape the eggplant flesh from the skin and add the flesh to the puréed beans in the food processor, along with the lemon juice, chopped mint, and oregano. Pulse briefly to form a chunky dip. Adjust the seasoning with more salt, pepper, or lemon juice to taste. Serve sprinkled with the pine nuts and mint leaves, with the toasted pita crisps on the side for dipping.
This is also great with crudités, especially bell peppers and fennel.