Preheat oven to 400° F. Lightly grease a 13x9-inch baking dish; set aside.
Place a large colander in the kitchen sink and add the broccoli florets. Carefully pour half a kettle of boiling water over the broccoli. Toss gently, then pour the remaining boiling water over the broccoli. Repeat and drain while preparing the rice.
Melt 3 tablespoons butter in a Dutch oven over medium heat. Add the onion, salt and black pepper and cook until softened, about 5 minutes. Add the rice and garlic and cook, stirring constantly, about 2 minutes. Add the chicken broth and milk. Increase the heat to medium-high and bring to a boil. Reduce the heat to medium-low and gently simmer, uncovered, stirring frequently until the rice is tender, about 10 to15 minutes. There should be some milky liquid remaining. If the rice absorbed all the milk and broth, add ¼ to ½ cup additional milk to the pot.
Remove from the heat and add the Dijon mustard, cayenne, 2 cups cheddar cheese and ½ cup Parmesan. Stir to combine. Fold in the diced chicken and drained broccoli. Mix gently to combine. Pour the mixture into the prepared baking dish and top with the remaining ½ cup cheddar. (At this point the casserole can be covered and refrigerated for up to 1 day.)