Heat vegetable oil in a large saucepan over medium-high heat. Add chicken, and cook until it begins to brown on the edges, but is still pink in the center, about 7 minutes. Stir in garlic, red pepper flakes, black pepper, and salt. Cook and stir until the garlic has begun to soften, and the chicken has become firm, about 3 minutes.

Stir in potatoes, carrots, and onion. Whisk hot pepper paste into soy sauce until dissolved, then pour into saucepan along with the water. Bring to a simmer, then reduce heat to medium-low, cover, and simmer until the chicken is no longer pink in the center, and the potatoes are tender, about 25 minutes. Stir occasionally.