Brush eggplant slices with 2 tablespoons of the oil.
In a large oven-going skillet brown eggplant slices, in batches, turning once. Transfer browned slices to a plate.
In the same skillet heat remaining 1 tablespoon oil over medium heat. Add mushrooms, shallots, and garlic. Cook 6 to 8 minutes or until mushrooms are tender and lightly browned, stirring occasionally. Stir in the next six ingredients (through oregano) and 3/4 cup of the cheese until combined. Top evenly with eggplant slices, overlapping as needed. Sprinkle with thyme, salt, pepper, and the remaining cheese. Cover and cook about 7 minutes or until heated through and cheese is melted.
If desired, preheat broiler. Uncover skillet and broil 3 to 4 inches from heat about 1 minute or until cheese is golden and bubbly. If desired, garnish with oregano.