Heat 2 tablespoons canola oil and sesame oil in a large skillet over medium-high heat. Cook and stir chicken and garlic in hot oil until fragrant, about 1 minute. Stir chile paste into chicken mixture; cook and stir until chicken is completely browned, 3 to 4 minutes. Add soy sauce and simmer for 2 minutes. Pour chicken and sauce into a bowl.
Heat 1 tablespoon canola oil in the skillet over medium-high heat; cook and stir cabbage, onion, carrots, and salt in hot oil until cabbage is wilted, 3 to 4 minutes.
Stir the chicken mixture into the cabbage mixture. Add noodles; cook and stir until noodles are hot and chicken is no longer pink inside, 3 to 4 minutes. Garnish with pickled ginger.