Make spicy paste. Add onion, jalapeno, garlic, gochujang, gochukaru, soy sauce, and mirin to a food processor. Pulse until a rough paste forms. (Or, mince all the vegetables finely and add to a large bowl with gochujang, gochukaru, soy sauce, and mirin.
Combine chicken + spicy paste. Add chicken to a large pot. Scrape spicy paste over the chicken. Add 1 cup water.
Cook. Heat pot to medium high heat and bring to boil. Lower heat, cover with a lid, and simmer on low until chicken is cooked through and tender, about 25-30 minutes. Keep an eye on the chicken, and scrape the bottom with a spatula from time to time, as the bottom can burn.
Add potatoes and carrots. Remove lid and add potatoes and carrots. Using a spoon, mix the potatoes into the braising liquid so that the potatoes are mostly submerged. Cover and simmer until potatoes are soft, about 10 more minutes.
Reduce sauce. Remove lid and check the braising liquid. If it looks watery, simmer with the lid off for a few more minutes until thickened. If it looks too thick, add a little water (2-4 Tbsp). The consistency should be similar to a thick gravy.
Serve. Garnish with sesame seeds and green onion. Serve hot with rice and kimchi.
