Peel and finely chop the shallots and garlic. Deseed the green chillies and finely chop. Fry the onion, garlic and chillies in olive oil. Let cool, then combine with the ground meat, egg yolks, cumin and salt. Slice the eggplants in half lengthwise, and make a few incisions in the cut surface. Peel and grate the garlic clove and mix with the olive oil.
Brush the eggplant with some of the oil mixture. Grill them over indirect heat until tender. Scoop out the flesh and combine with the remaining olive oil mixture. Blanch and peel the tomatoes. Halve them, remove the seeds and chop into small cubes. Shred the oregano and parsley leaves. Combine the tomatoes, herbs and lemon with the eggplant. Season to taste with paprika and salt.
Place the olives on some paper towels on a plate and dry them in the microwave for 5 min. at 900w. Coarsely chop or crush them. Peel and thinly slice the red onion. Toast the bread slices on the barbecue or toaster. Coarsely chop the mint. Form the meat into 4 patties and grill until golden, approx. 2 min. on each side. Cover with the cheese and let melt.
Top the bread with arugula, onions and meat. Sprinkle with the chopped olives and mint. Serve with the eggplant-tomato cream.