Remove tofu from packaging and drain from packing liquid. Wrap in paper towels, set on a wire rack or colander and weigh down with a saucer for about 15 to 20 minutes to extract moisture. Cut into 1-inch cubes.
In deep, wide pan, heat about 2-inches deep of oil until very hot but not smoking. Add tofu in batches and cook, turning as needed, until golden and crisp. With a slotted spoon, remove from pan and drain on paper towels.
In a bowl, soak mushrooms in 1 ½ cups of warm water for about 15 to 20 minutes or until softened. With hands, squeeze mushrooms to extract liquid. Reserve 1 cup of the soaking liquid. Trim mushroom stems and cut caps into halves.
Fill a bowl halfway with ice and enough water to cover ice. Add ½ teaspoon salt for each quart of water. Set aside.
In a sauce pan, add 3 cups of salted water and bring to a boil. Add carrots and cook for about 1 minute or until half done. With a slotted spoon, remove from pan and plunge into bowl of ice bath. Add broccoli to the boiling water and cook for about 2 to 3 minutes or until half done. With a slotted spoon, remove from pan and plunge into bowl of ice bath. Add cabbage to the boiling water and cook for about 30 seconds or until half done. With a slotted spoon, remove from pan and plunge into bowl of ice bath. Reserve 2 cups of the poaching liquid (the one used to blanch vegetables). Drain vegetables from the ice bath when they are cold.
In a wok or wide skillet or medium heat, heat oil. Add onions and garlic and cook until softened.
In a bowl, combine reserved 1 cup of poaching liquid (from par-boiling vegetables), 1 cup of the soaking liquid (from soaking mushrooms), and cornstarch. Stir until cornstarch is well dissolved. Add to wok and bring to a simmer until slightly thickened.
Add parboiled vegetables, fried tofu, and Shiitake mushrooms, stirring to combine, and cook for about 3 to 5 minutes or until vegetables are heated through. Season with salt and pepper to taste. Serve hot.