Pat chicken thighs completely dry with paper towels, then season both sides generously with salt and black pepper. Set aside at room temperature for about 10 minutes while you prep the vegetables.
Cut potatoes into 1.5-inch chunks, carrots into 1-inch diagonal slices, and onion into large wedges. Mince the garlic, grate the ginger, and if using, grate the peeled apple on the fine side of a box grater; set aside.
Heat oil in a large Dutch oven over medium-high heat until shimmering, then sear chicken thighs skin-side down for 4–5 minutes until deeply golden; flip and sear 3–4 more minutes. Remove and set aside — the chicken doesn't need to be cooked through.
Reduce heat to medium, add onion wedges to the drippings, and cook 5–7 minutes until softened and golden at the edges. Add garlic and ginger and stir for 60 seconds until fragrant.
Stir in potatoes and carrots, then pour in chicken broth and water, scraping up any browned bits from the bottom. Add soy sauce, Worcestershire sauce, honey, and grated apple if using, then nestle the browned chicken back in skin-side up.
Bring to a boil over medium-high heat, skim any foam from the surface, then reduce to medium-low, cover with lid slightly ajar, and simmer 25–30 minutes until potatoes are fork-tender and chicken reaches 165°F (74°C).
Remove pot from heat entirely, then add curry roux pieces one or two at a time, stirring after each addition until fully dissolved with no lumps remaining.
Return pot to medium-low heat and simmer uncovered, stirring frequently, for 8–10 minutes until the sauce is thick, glossy, and coats the back of a spoon. Taste and adjust with soy sauce, honey, or salt as needed.
Spoon steamed rice into bowls and ladle curry alongside or over the rice. Garnish with sliced green onions and fukujinzuke pickled relish, and serve immediately.