Preheat the oven to 450 degrees. Poke the eggplant with the tip of a sharp knife in a few places and set on a rack-covered baking sheet; roast until tender, about 30 minutes. Finely chop 2 large cloves garlic and reserve. Cut the end of the remaining head of garlic to expose the cloves; drizzle with EVOO and season with salt and pepper. Finely chop the rosemary and press the garlic head into it to coat. Wrap the garlic head in foil and roast with the eggplant until soft and golden, about 35 minutes. Let the eggplant and garlic cool.
Squeeze the garlic into a food processor. Cut open the eggplant and scoop the flesh into the food processor; discard the skin. Puree the garlic and eggplant with the mint and lemon juice; season with salt and pepper.
Combine the lamb with the reserved garlic, the parsley, chives, paprika or harissa, cumin and coriander; season with salt and pepper. Form the mixture into 4 patties, making them slightly thinner at the center than at the edges. In a large nonstick or cast-iron skillet, cook the burgers in a drizzle of EVOO over medium-high heat, turning once, for 6 minutes for rare or 8 minutes for medium. Top with the cheese during the last minute of cooking.
Spread the eggplant puree on the top bread halves (reserve any leftover puree to have as a snack on charred bread). Layer the bottom bread halves with the lettuce, burgers, tomatoes and onions. Cover with the bread tops.