Slice the eggplant into ¼-inch rounds and place in a colander. Sprinkle the slices evenly with 3 teaspoons of salt. Let sit for 30 minutes to drain. Lightly rinse and pat dry with a paper towel.

While the eggplant is draining, preheat 2 tablespoons of olive oil in a large skillet over high heat. When the oil shimmers, add the beef and cook until nicely browned, about 10 minutes. Once browned, add the tomato paste, 2 teaspoons of salt, and pepper. Cook until the tomato paste turns a dark brick red, about 5 minutes. Remove from the heat, and stir in the mint. Set aside to cool slightly.

Preheat the oven to broil. Lightly coat the bottom of a 9x13-inch baking dish with olive oil. Working in batches, dip the eggplant slices into the oil on both sides. Arrange the eggplant in the baking dish in a single layer. Working in batches, broil the eggplant until lightly browned, about 4 minutes. Transfer to a plate, and repeat with remaining slices. 

Preheat the oven to 350 degrees and assemble the ingriye: Line the bottom of the baking dish with half of the eggplant, browned side down. Then spread the browned beef mixture in an even layer, followed by the sliced bell peppers, and ⅔ of the tomatoes. Shingle the remaining half of the eggplant on top, browned side up. Spoon the remaining ⅓ of the tomatoes on top.

Cover with tinfoil and bake at 350°F for 1 hour. After an hour, uncover and add the lemon juice. Cover and bake for another 30 minutes. Remove from the oven, uncover and let cool for 15-20 minutes. Serve with rice and salad, or on its own.