Soak cheese in a bowl of water for 30 minutes. Drain.
Place eggplant in a colander; sprinkle with 1/2 teaspoon salt. Toss well. Drain 30 minutes. Rinse and pat dry with paper towels.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add onion; sauté 15 minutes or until golden brown, stirring frequently. Add 1/4 teaspoon salt, bulgur, boiling water, and pepper; bring to a boil. Cover, reduce heat to medium low, and simmer 15 minutes or until liquid is absorbed.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add eggplant; sauté 10 minutes or until browned. Combine cheese, bulgur mixture, eggplant, and 2 teaspoons oil in a Dutch oven. Toss well to coat; place over medium heat 5 minutes or until cheese is soft. Serve hot.