Preheat oven to 400°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.
Whisk in milk and broth until combined.
Cook, whisking often, until mixture comes to a simmer and has thickened, about 7 minutes.
Remove from heat. Stir in herbs, salt, and pepper.
Gently stir in rice and chicken. Taste for seasonings; add salt ¼ teaspoon at a time to reach desired flavor.
Transfer to an 11x7-inch casserole dish.
In a medium bowl, stir together breadcrumbs and remaining 2 tablespoons butter.
Sprinkle breadcrumbs over casserole. Bake until golden and bubbly, about 15 minutes.