Grease two individual (or one larger casserole) dishes with butter and set aside.
In a large pan, melt butter over medium heat. Add the onion and carrot and cook, stirring occasionally until softened.
Add the garlic and cook until just fragrant, 30 seconds to 1 minute.
Crumble in the ground beef and cook, stirring occasionally until browned.
Stir in the curry powder and cook until fragrant. Season with the salt and pepper.
Pour in the chicken broth (the liquid should be enough to cover the beef), ketchup, and Worcestershire sauce, and bring to a boil. Reduce to a simmer and cook, stirring often and removing any foam that develops on the top, until the mixture has been reduced to a thickened sauce.
Adjust seasonings and remove from heat.
Divide the cooked rice between the two individual gratin dishes in an even layer.
Cover each with the curry beef mixture in an even layer.
Shred the Gruyère cheese and grate the Parmesan cheese. Divide across the top of each dish. Sprinkle with the Panko breadcrumbs.
Place the gratin dishes on a baking sheet and put under the broiler. Broil until the cheese is golden and bubbly.
Remove from heat and serve topped with the parsley if desired.