In a medium bowl, place four layers paper towel, add tofu and cover with four more layers of paper towel, pressing lightly to extract liquid from the tofu. Let tofu stand for 10 minutes to drain, then cut into 1-inch cubes.
Meanwhile, in 2-cup liquid measuring cup, with fork or wire whisk, combine 1/2 c. water, broth, soy sauce, brown sugar, and cornstarch, stirring until brown sugar and cornstarch are dissolved. Set aside.
In deep 12-inch skillet or wok, heat 1 Tbl. oil over medium-high heat until hot. Add eggplant and remaining 1/3 c. water, cover and cook until eggplant is tender, 7 to 10 minutes, stirring occasionally. Transfer eggplant to a small bowl, set aside.
Add remaining 1 Tbl. oil to skillet and heat until hot. Add mushrooms and tofu, cook until tofu is lightly brown, about 5 minutes. Stir in ginger, garlic, and half of the green onions, cook 1 minute, stirring. Add bok choy and cook until vegetables are lightly browned, about 4 minutes longer.
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Stir broth mixture, add to tofu mixture with eggplant. Heat to boiling over medium-high heat, reduce heat to low and simmer 1 minute, stirring. Sprinkle with remaining green onion before serving.
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