Halve the eggplants lengthwise and cut out the flesh, leaving 1/4-inch shells. Cut the flesh into 1/2-inch dice. Salt the eggplant shells and let stand for 30 minutes. Pat the shells dry.
Preheat the oven to 350°. Rub the eggplants with oil; set them on a rimmed baking sheet, cut side down. Add 1/4 cup of water, cover with foil and bake for 45 minutes.
Meanwhile, in a skillet, heat 2 tablespoons of olive oil. Add the mushrooms. Season with salt and pepper, cover and cook over moderate heat until tender; transfer to a bowl. Heat 1 tablespoon of olive oil in the skillet. Add the diced eggplant. Season with salt and pepper, cover and cook until tender and browned, 3 minutes; add to the mushrooms.
Add the garlic, onion and 2 tablespoons of olive oil to the skillet. Cover and cook, until softened. Add the cumin and 1 tablespoon of the butter and stir until fragrant, 1 minute; add to the mushrooms. Stir in the wine-soaked bread, cheese and broth and season the filling with salt and pepper.
Increase the oven temperature to 425°. Turn the eggplant shells cut side up and fill with the bread stuffing. Dot the tops with the remaining 1 tablespoon of butter and bake in the upper third of the oven for 10 minutes.
Preheat the broiler. Broil the eggplant 4 inches from the heat until browned, 2 minutes. Top with the parsley and serve.