Cut the eggplants into slices lengthwise with a thickness of about 0.5″ (1 cm) to 2″ (5 cm). Salt and leave for a few minutes in an inclined tray to drain the bitter juice.
Put about 1/2 cup of oil in a baking dish and arrange one layer of eggplant. Place in a heated oven. Whilst the eggplants bake, prepare a sauce from the mince and tomato paste. Sauté the crushed garlic in oil, add the minced meat and braise for 5 minutes.
Add the tomato puree and leave it until the water boils off. When the eggplants brown, (after about 15-20 minutes) remove and add a few tablespoons of tomato sauce and grated cheese on each eggplant. Return to the oven for about 10 more minutes until the cheese is browned.