Preheat oven to 400 degrees F (200 degrees C). Grease a 3-quart casserole dish.
Melt butter in a saucepan over medium heat. Mix in flour and tarragon; cook for about 1 minute. Pour in 1 cup chicken broth and heavy cream gradually; stir continuously until smooth and thick, 6 to 8 minutes. Mix chicken breast, salt, and white pepper into sauce.
Fill casserole dish with cooked rice. Spread broccoli on top. Pour remaining 1/4 cup chicken broth and sauce over broccoli. Sprinkle cashews evenly on top.
Bake in the preheated oven until hot and bubbly, about 20 minutes.