In small bowl, combine the shiitakes and hot water to cover. Let stand until softened, about 30 minutes.
Meanwhile, drain the tofu and pat dry with paper towels. Cut into 3/4-inch cubes; pat dry again. In medium bowl, combine tofu, 1 tablespoon soy sauce and five-spice powder; toss gently to coat. Cover and set in refrigerator.
Bring a large saucepan of lightly salted water to a boil. Cut broccoli florets into 1 1/2-inch pieces. Peel tender portion of stems and cut into 1/2-inch-thick slices. (You should have about 4 cups.) Add broccoli to boiling water and cook just until bright green and crisp-tender, about 3 to 4 minutes. Drain, rinse under cold running water and drain again. Set aside.
In small bowl, mix together cornstarch and 1 tablespoon cold water; set aside. Remove softened shitakes from soaking water. Strain soaking water through a sieve into medium bowl; stir in broth, brown sugar and remaining 2 tablespoons soy sauce. Set aside. Trim and discard shitake stems(Save and use in soup!); slice caps and set aside.
In large nonstick skillet, heat 2 teaspoons olive oil over medium-high heat. Add tofu mixture and cook, shaking pan and tossing tofu, until lightly browned, 3 to 4 minutes. Transfer to plate and set aside.
In same skillet, heat remaining 2 teaspoons olive oil over medium-high heat. Add scallions, garlic, ginger, chili paste, anise seeds and reserved shiitakes and stir-fry until fragrant, 30 to 60 seconds. Add cinnamon stick and reserved broth mixture; bring to a boil and cook 5 minutes. Stir cornstarch mixture and add to skillet; cook, stirring, until liquid has thickened, about 30 seconds. Add carrots and reserved broccoli and cook, stirring often, until carrots are tender, about 3 minutes. Add reserved tofu and toss gently to coat. Discard cinnamon stick and serve. Enjoy!