Rinse and dry eggplant; slice crosswise into 1/4-inch-thick rounds, discarding ends. Brush both sides of rounds lightly with oil and place in a single layer on two 12- by 15-inch baking sheets. Bake in a 400° regular or convection oven until browned and tender when pierced, 10 to 15 minutes.
Meanwhile, cut or tear bread into 1/2-inch chunks. Put in a food processor or blender and whirl into coarse crumbs; you should have 1 cup. In a 10- to 12-inch nonstick frying pan over medium-high heat, stir crumbs in 1/2 tablespoon olive oil until lightly browned and crisp, about 5 minutes. Pour from pan.
In the same pan over medium-high heat, stir lamb until crumbled and no longer pink, about 5 minutes. With a slotted spoon, transfer lamb to a bowl. Drain all but 1 teaspoon fat from pan. Add onion, mushrooms, and garlic and stir occasionally until onion is limp and mushrooms are browned, about 8 minutes. Add tomato sauce, wine, mustard, basil, oregano, salt, pepper, and the browned lamb. Bring to a simmer and cook, stirring often, to blend flavors, about 10 minutes.
Arrange half the eggplant slices in a single layer over the bottom of a shallow 2- to 3-quart baking dish; top with half the lamb mixture. Repeat layers of eggplant and lamb mixture. Sprinkle top with bread crumbs and feta cheese.
Bake in a 350° regular or convection oven until browned and bubbling, 20 to 25 minutes. Sprinkle with parsley. Scoop out servings with a large spoon.