In a saucepan, melt the butter over medium heat. Whisk in the all-purpose flour to create a smooth roux. Cook the roux, stirring constantly, for 3-5 minutes, until it turns light golden in color.
Add the diced onion, carrot, and potato to the saucepan. Sauté the vegetables for 5-7 minutes, or until they start to soften.
Gradually pour in the chicken or beef broth, whisking continuously to prevent any lumps from forming.
Stir in the soy sauce, Worcestershire sauce, brown sugar, and curry powder. Bring the mixture to a simmer and let it cook for 15-20 minutes, stirring occasionally, until the sauce has thickened to your desired consistency.
Taste the sauce and adjust the seasoning as needed, adding more soy sauce for saltiness, brown sugar for sweetness, or curry powder for a more pronounced flavor.
If the sauce is too thick, you can thin it out by adding more broth. Conversely, if it's too thin, let it simmer for a bit longer to allow it to reduce and thicken.
Serve the katsu curry sauce warm, drizzled over your favorite protein, such as tonkatsu (fried pork cutlets), or alongside steamed rice.