Set the oven to 400°F (200°C) and grease a large baking dish with a bit of butter.

In a wide skillet, melt 2 tablespoons of butter over medium heat. Toss in the onion and cook until translucent, then stir in the garlic and let it cook for about a minute until fragrant.

Drop the remaining butter into the same skillet. Once melted, sprinkle in the flour and whisk continuously for roughly two minutes, letting the roux turn a light golden color.

Pour the chicken broth in gradually, whisking the whole time so no lumps form, then repeat the process with the milk. Keep the heat on low-medium and let the sauce simmer until it turns into a smooth, gravy-like texture.

In a large mixing bowl, combine the shredded chicken, broccoli, cooked rice, half of the cheddar, and the prepared sauce. Fold everything together gently until evenly coated, then transfer the mixture into your prepared baking dish.

In a separate small bowl, stir together the panko crumbs, melted butter, and the remaining cheddar cheese. Scatter this topping evenly across the casserole.

Slide the dish into the oven and bake for 20-25 minutes, until the surface turns golden brown and slightly crisp.