Prepare all the ingredients before you start cooking: thinly slice the onion, mince the garlic, grate the ginger, cut the chicken thighs into bite-sized pieces, slice the carrots into rounds, and cut the potatoes into chunks. Break the Japanese curry roux into smaller blocks so they dissolve easily.
Heat the vegetable oil in a large pot over medium heat. Add the sliced onion and cook, stirring often, for about 5 to 7 minutes until the onion softens and starts to turn lightly golden around the edges.
Add the minced garlic and grated ginger to the pot. Cook for about 1 minute, stirring constantly, until fragrant but not browned.
Add the chicken pieces to the pot in an even layer. Season lightly with a pinch of salt and black pepper. Cook for 4 to 5 minutes, stirring occasionally, until the chicken is no longer pink on the outside.
Add the sliced carrots and potato chunks to the pot. Stir well to coat the vegetables with the oil, aromatics, and chicken juices. Cook for 2 to 3 minutes, stirring occasionally.
Pour in the chicken broth and water. Stir to combine, scraping the bottom of the pot to release any browned bits. Bring the mixture up to a gentle boil over medium-high heat.
Once it reaches a boil, reduce the heat to low, skim off any foam that rises to the surface if needed, and cover the pot with a lid slightly ajar. Simmer gently for about 15 to 20 minutes, or until the potatoes and carrots are tender when pierced with a fork.
Turn the heat down to very low. Add the Japanese curry roux blocks to the pot a few pieces at a time, stirring constantly as they melt into the liquid. Continue stirring until all the roux has fully dissolved and the sauce begins to thicken.
Stir in the soy sauce and sugar. Let the curry simmer on very low heat for another 5 to 10 minutes, stirring occasionally, until the sauce is glossy and thick enough to coat the back of a spoon. Taste and adjust the seasoning with a pinch of salt and black pepper if needed.
To serve, spoon hot cooked Japanese short-grain rice into shallow bowls or plates. Ladle the hot Japanese curry over one side of the rice, leaving part of the rice visible. Sprinkle with chopped green onion for garnish if using, and serve immediately while hot.