Bring a large pot of lightly salted water to a boil. Cook udon in boiling water, stirring occasionally, until noodles are tender yet firm to the bite, 10 to 12 minutes. Drain noodles and rinse with cold water.

Meanwhile, warm olive oil in large saucepan or wok over medium heat. Add chicken and sauté until cooked through, 3 to 4 minutes. Add onion, bell pepper, cabbage, carrots, and garlic. Cook for 3 to 4 minutes, allowing vegetables to maintain most of their crispness.

Pour soy sauce over vegetables and chicken; sauté 1 to 2 minutes more. Add udon noodles and toss well. Mix in gojuchang and ketchup, thoroughly combining to mix all the flavors. Season with salt and pepper.