Butter or grease a 9 × 9-inch casserole.
In a medium saucepan, melt butter; add the flour and whisk until blended. Add the milk and then the broth in a stream, whisking constantly. Keep whisking and cooking until the white sauce thickens; remove from heat.
In a large bowl (or the saucepan if it’s large enough), gently mix together all ingredients except cracker crumbs. Mix as little as possible so that the chicken doesn’t shred. (Overmixing will do that.) Spoon mixture into prepared dish, top with the cracker crumbs, and bake in a preheated 350° oven for 35-45 minutes, or until casserole is bubbly and cooked through.