Combine 2 tablespoons sesame oil, soy sauce and crushed red pepper in a large bowl. Add chicken, toss to coat, and set aside.

Bring a large pot of water to a boil. Cook noodles in the boiling water according to package directions. Drain, transfer to a large bowl and toss with the remaining 2 teaspoons sesame oil and chile-garlic sauce. Set aside.

Heat 1 tablespoon vegetable oil in a large skillet or wok over medium-high heat until shimmering. Add mushrooms and cook until juices are released and the mushrooms are golden and tender, 4 to 5 minutes. Stir in garlic, ginger and scallions and cook until fragrant, 1 minute. Transfer to a medium bowl.

Add 1 tablespoon vegetable oil to the pan and heat over medium-high heat until shimmering. Add broccoli, bell pepper, and snow peas. Cook until tender-crisp, 2 to 3 minutes. Stir in cabbage and cook until wilted, 2 minutes. Transfer to the bowl with the mushroom mixture.

Heat the remaining 1 tablespoon vegetable oil in the pan over medium-high heat. Add the reserved chicken, leaving the sauce in the bowl. Sauté the chicken until cooked through, 2 to 3 minutes. Meanwhile, add broth and cornstarch to the reserved sauce and whisk to combine. Add the sauce to the pan with the chicken; simmer until thickened, 1 to 2 minutes.

Add the vegetable mixture to the pan and toss to combine. Transfer the chicken and vegetables to the bowl with the noodles and toss to combine.