Cook the chicken: Season chicken breasts with salt and pepper. Cook them in a skillet over medium heat for about 7-8 minutes on each side, until fully cooked. Shred the chicken and set aside.
Make the béchamel sauce: In a saucepan, melt butter over medium heat. Stir in the flour and whisk for about 1-2 minutes. Gradually add the chicken broth and milk, whisking constantly until the sauce thickens. Season with nutmeg, salt, and pepper.
Assemble the casserole: In a 9x13-inch baking dish, layer the cooked chicken and steamed broccoli. Pour the béchamel sauce over the top and spread evenly. Sprinkle shredded cheese and breadcrumbs on top (if using).
Bake: Preheat the oven to 375°F (190°C). Bake the casserole for 25-30 minutes, until the top is golden and bubbly.
Rest and serve: Let the casserole rest for 5 minutes before serving.