Blanch the broccoli in boiling salted water for 2 minutes, then drain and cool briefly.
Melt 2 tablespoons of butter in a large saucepan. Cook the onion until soft, add the garlic, then whisk in the flour.
Gradually whisk in the milk and broth. Simmer until thick enough to coat a spoon, about 4 minutes.
Remove from the heat and stir in 1 1/2 cups of cheddar, mustard, salt, and pepper.
Fold the sauce, rice, and broccoli together in a greased 9-by-13-inch baking dish.
Melt the remaining butter and toss it with the breadcrumbs and remaining cheddar. Sprinkle over the casserole.
Bake at 375°F for 25 to 30 minutes, until bubbling at the edges and golden on top. Rest 10 minutes before serving.
Notes
Model-authored fallback: the source page was used for the dish identity; this coherent from-scratch recipe supplies the canonical quantities and method.