Preheat oven to 375°F. Salt the eggplant slices on both sides.
Heat 1 tsp of the coconut oil in a heavy pan over medium heat and sear the eggplant until golden brown. Remove to plate.
Lightly oil a 2.5qt casserole dish with coconut oil and lay the eggplant in an overlapping pattern. (See pictures for visual help).
Heat remaining coconut oil in pan over medium heat and sauté the mushrooms and onions until cooked through (about 10 minutes), stirring often. Add the beef and cook for another 10 minutes, breaking up the meat as it cooks.
Stir in the tomatoes, tomato paste and herbs and cook another 2-3 minutes. Drain the meat mixture and season with salt and pepper, if desired.
Pour over eggplant layer and distribute evenly across. Top with overlapping layer of thinly sliced potatoes.
In a small bowl, mix together the yogurt, eggs (and egg whites, if using), salt and pepper and feta cheese. Pour over the potato layer and bake for 45-60 minutes (place on baking tray if your casserole dish is very full). Let stand 15-20 minutes before slicing.