In a bowl, soak the potato cubes in cold, salted water (while preparing the other ingredients). This will help to remove excess starch, and prevent the potatoes from browning.
In a heavy bottom pot, or deep skillet, heat two tablespoons of vegetable oil. When the oil is hot, add in the onion. Sauté over medium-high heat until the onions are softened.
Add in small splashes of water if it gets too dry. Cook for about 10 minutes.
Next, add in the garlic and ginger, and cook for about a minute. Then mix in the Japanese curry paste, and cook for another 20-30 seconds until fragrant.
Add in the carrots and Japanese turnips (or cauliflower), and sauté for 3-4 minutes, or until the vegetables are just tender.
Stir in the broth and potato cubes, and heat to a simmer. Peel the apple, and grate it into the curry. Add in the ketchup, Worcestershire sauce, flour slurry, and miso paste. Stir well to fully combine.
Continue to simmer for 25-30 minutes, or until potatoes are fork-tender. Add in the sweet peas (or edamame), and simmer for another 5-8 minutes.
Have a taste and adjust any seasonings if needed. The curry should thicken to a stew-like consistency. Serve the curry with Japanese rice or brown rice.