Preheat oven to 350°. Spray a 13x9-inch baking dish with cooking spray.
In a large Dutch oven, melt butter over medium heat. Add onion and garlic; cook, stirring occasionally, until tender, about 5 minutes. Whisk in flour; cook, whisking constantly, for 2 minutes. Gradually whisk in broth; bring to a boil. Reduce heat to medium- low; cook, whisking constantly, until thickened, about 2 minutes.
Remove from heat; whisk in cream cheese, poppy seeds, thyme, mustard, salt, lemon zest, and pepper until smooth. Stir in chicken, rice, and broccoli; transfer to prepared pan.
In a small bowl, stir together crackers and melted butter; sprinkle onto chicken mixture.
Bake until golden brown and bubbly, 25 to 30 minutes, covering with foil to prevent excess browning, if necessary. Let stand for 10 minutes before serving.
Recipe by Paula Deen Magazine at https://pauladeenmagazine.com/chicken-poppy-seed-and-rice-casserole/