Combine all the sauce ingredients in a bowl and set aside.
Heat the oil in a large frying pan on high heat then add the sliced onion and minced garlic, cooking until they start to turn golden, about 1 and a half minutes.
Add the chicken pieces and sauté just until the exterior shifts from pink to white.
Pour in the prepared sauce and continue cooking for another minute until the chicken takes on a deep caramelized color.
Stir in the carrot matchsticks and sliced capsicum, cooking everything together for an additional minute to soften the veggies slightly.
Push the cooked chicken and vegetables to one side of the pan, add the water, and submerge the noodles into the simmering liquid.
Once you notice bubbles forming around the edges, let the noodles cook for 45 seconds, then turn them gently.
Wait another 30 seconds before loosening the noodles with a fork, then mix them thoroughly with the chicken and vegetables.
Add the shredded cabbage and keep tossing the mixture for about one minute until the sauce thickens and coats all ingredients, and the noodles are tender.
Serve the ramen hot, topped with finely sliced green onions or shallots for a fresh garnish.