Preheat oven to 350°F.
Melt 2 tablespoons butter in a Dutch oven over medium heat.
Add onion and garlic; cook, stirring occasionally, until softened, about 5 minutes.
Add another 4 tablespoons butter; let met.
Stir flour into butter and onions. Cook, stirring, 2 minutes. Stir in stock and milk until combined.
Cook, stirring oven, until mixture comes to a simmer and has slightly thickened, about 5 minutes.
Remove from heat. Stir in mustard and 1 ½ cups cheese until smooth. Gently stir in broccoli and rice.
Taste for seasonings; add salt ¼ teaspoon at a time to reach desired flavor. Stir in pepper.
Transfer to a 13x9-inch casserole dish.
In a medium bowl, stir together breadcrumbs and remaining 2 tablespoons butter.
Sprinkle breadcrumbs over casserole; top with remaining 1 cup cheese.
Bake until golden and bubbly, about 30 minutes.