Pat tofu dry and toss gently with cornstarch to coat. In a wok or large non-stick skillet, heat 4 teaspoons vegetable oil over medium-high heat; cook tofu, turning occasionally, until crisp and golden, 8 to 10 minutes. Transfer to paper towel-lined plate to drain. Set aside.
In same wok, heat remaining 2 teaspoons vegetable oil over medium-high heat; stir-fry mushrooms, garlic and ginger until mushrooms are beginning to soften, about 2 minutes. Add broccoli; stir-fry until tender-crisp, about 3 minutes. Stir in hoisin sauce, soy sauce, chili sauce and ¾ cup water; bring to boil.
Reduce heat to low; add noodles, tossing gently to coat. Simmer, stirring gently, until sauce is slightly thickened, about 2 minutes. Add tofu, green onions and sesame oil; cook, tossing to coat, for 1 minute.