Melt the butter in a large saucepan then add the onion and cook for 4-5 minutes until softened. Add the garlic and cook for another minute, stirring frequently. Then add the flour and cook for 1-2 minutes, stirring frequently, until lightly browned.
While the onion cooks, steam the broccoli until tender.
Slowly pour in the broth and milk while whisking constantly and continue whisking for about a minute. Then stir in the salt and pepper and bring to a simmer and cook until thickened, 3 to 4 minutes.
Remove from the heat and add the cooked chicken, rice, and broccoli and cheese then stir to combine. Transfer the casserole mixture to an 8 or 9 inch square baking dish then top with the crushed crackers.
Bake for 20-25 minutes until the casserole is bubbly and lightly browned on top. Allow to cool for about 5 minutes before serving.