Preheat oven to 375°F (190°C). Grease a 9×13-inch baking dish.
In a skillet, melt butter over medium heat. Add onion, garlic, celery, and mushrooms. Sauté until soft.
Stir in flour to make a roux. Cook 1–2 minutes.
Gradually whisk in milk. Add Dijon and Worcestershire. Simmer until thickened.
Stir in cheddar, Parmesan, salt, and pepper. Remove from heat.
In a large bowl, combine rice, chicken, carrots, and sauce. Mix well.
Pour mixture into the baking dish. Spread evenly.
Toss panko with melted butter. Sprinkle over the top.
Bake for 30–35 minutes or until golden and bubbly.
Garnish with parsley. Let rest 10 minutes before serving.