Heat oil in a skillet over medium heat. Mix in garlic and diced hot pepper. Cook and stir until golden. Remove from heat and allow to cool.
Bring a large pot of lightly salted water to a boil. Place noodles in pot and cook according to directions, until al dente. Drain and transfer to a large bowl. Set aside 2 tablespoon garlic oil and toss remainder with pasta to coat. Season pasta with salt.
Bring a pot of water to a boil. Immerse broccoli, cabbage, carrots, cauliflower, hot pepper, red pepper, and green onions in water for about 30 seconds. Drain and set aside.
Heat reserved garlic oil in skillet over medium heat. Mix in chicken and 2 tablespoons soy sauce. Cook until chicken juices run clear. Mix in vegetables and remaining soy sauce. Season with salt and pepper. Toss with noodles, and garnish with cilantro and lime to serve.