Heat the oven to 350°F. Combine the breadcrumbs and half of the cheese. Set aside.
Heat the stock and milk in a medium saucepan over medium-low heat until steaming. Keep warm while you prepare the vegetables.
Heat the butter in a large ovenproof casserole or skillet over medium heat. Add the onion and celery. Cook until the onion is soft and translucent, 3 to 5 minutes. Add the garlic and cook until fragrant, 45 to 60 seconds. Add the flour and stir to combine. (The mixture will be dry.)
Whisk the stock into the flour and vegetables. Add the remaining cup of cheese and stir until smooth. Add the rice and stir to combine. Bring to a boil. Add the yogurt; stir until smooth. Add the chicken and cover with a tight-fitting lid and transfer to the oven.
Bake until the rice is tender and the liquid has been absorbed, about 40 minutes.
Once the rice is tender and the liquid has been absorbed, remove the casserole from the oven and stir in the broccoli.
Smooth the top and sprinkle with the reserved breadcrumb-cheese mixture. Return to the oven and bake until the topping is golden brown, about 15 minutes longer. Place the casserole on a rack to cool slightly before serving.