Chop eggplant in half, then in slices. Using 1 T oil, lightly brush eggplant slices on both sides. Set aside.
Heat 1-1/2 T oil in a large skillet over medium heat. Add half the garlic and cook, stirring for 30 seconds. Add meat, 1t oregano, salt, pepper, cinnamon and allspice. Cook, stirring and breaking up the meat, until no longer pink, 4-8 minutes.
Using a slotted spoon, remove meat and garlic from the pan, draining well, and transfer to a medium bowl. Add tomato puree (or crushed tomatoes), water and 1/3 C yogurt to the meat mixture; stir to combine.
Pour of any liquid in the pan and wipe the pan clean. Heat the remaining 1-1/2 T oil over medium heat. Add onion, remaining garlic, 1t oregano, and the eggplant; cook, stirring, until the eggplant has softened, 3-4 minutes.
Reduce heat to medium-low and add the meat mixture. Cover and cook, stirring occasionally, until the onion and eggplant are soft, 12-15 minutes. Sprinkle cheese and mint (or parsley) on top and stir to combine.