Heat the vegetable oil in a large pot or Dutch oven over medium-high heat. Add the chicken pieces and cook until lightly browned on all sides. Remove the chicken from the pot and set aside.
Add the chopped onion to the pot and sauté until softened and translucent, about 5-7 minutes. If the pot seems dry, add a little more oil.
Add the carrots and potatoes to the pot and stir for a minute to coat them in the residual oil and onion. Pour in the chicken broth or water, scraping up any browned bits from the bottom of the pot.
Bring the liquid to a boil, then reduce the heat to low, cover, and simmer for about 15-20 minutes, or until the vegetables are tender when pierced with a fork.
Return the browned chicken to the pot. Bring the mixture back to a gentle simmer.
Break the curry roux blocks into pieces and add them to the pot, stirring continuously until they are completely dissolved and the sauce begins to thicken. This usually takes about 5-10 minutes.
If using, stir in the frozen peas or bell peppers during the last few minutes of cooking.
Taste and adjust seasoning if necessary. If the curry is too thick, add a little more broth or water. If it's too thin, you can dissolve a little more roux in a separate bowl with some hot liquid before adding it to the pot.
Serve hot over steamed Japanese short-grain rice.