Arrange tofu in a single layer on a baking sheet lined with paper towels. Cover with more towels and another baking sheet; weight with a heavy skillet. Let drain until towels are soaked, about 15 minutes.
Meanwhile, make teriyaki sauce: Place ginger in a fine strainer; press to release juices into a bowl. Whisk in soy sauce, vinegar, sugar, cornstarch, and 1 cup water.
In a large nonstick skillet, heat 1 tablespoon oil over medium heat. Cook half the tofu until golden brown, 4 to 6 minutes per side. Transfer to a plate. Repeat with remaining tofu.
Heat remaining tablespoon oil in same skillet over medium heat. Cook mushrooms, stirring often, until browned, about 5 minutes. Add teriyaki sauce; bring to a boil. Cook until thickened, about 1 minute. Return tofu to pan; toss gently to coat. Serve with watercress.
In step four, the sauce should be just thick enough to coat the back of a spoon. If necessary, you can thin it by adding up to 1/4 cup more water. If you prefer, you can use bottled teriyaki sauce in this recipe. Omit the soy sauce, vinegar, and sugar in step two; add 1/4 cup of bottled sauce instead.