Warm the oil in a heavy pot over medium heat.
Sear the chicken until lightly browned, 4–5 minutes. Transfer to a plate.
Cook the onions in the same pot until soft and golden at the edges, 6–8 minutes.
Stir in garlic and ginger. Cook until fragrant, about 30 seconds.
Add carrots and potatoes. Toss to coat with the aromatics for 1 minute.
Pour in the broth and scrape up any browned bits. Return the chicken to the pot.
Bring to a boil, then reduce to a gentle simmer. Partially cover and cook until vegetables are tender, 15–18 minutes. Skim foam if needed.
Stir in grated apple, soy sauce, and Worcestershire. Season with salt and pepper.
Turn the heat to low. Whisk in the curry roux pieces a few at a time until dissolved.
Simmer gently, stirring often, until thick and glossy, 5–8 minutes. Fold in peas and butter, if using. Adjust seasoning.
Spoon the curry over bowls of hot rice and serve right away.