4 tablespoons olive oil, plus more for greasing
2/3 cup breadcrumbs
Kosher salt and freshly ground black pepper
4 boneless, skinless chicken breasts (about 2 pounds)
4 heads broccoli, cut into florets (about 6 cups)
2 cups grated Parmesan
1/4 cup unsalted butter
1/2 cup all-purpose flour
1/4 teaspoon freshly grated nutmeg
Pinch cayenne pepper
3 cups milk
2 cups shredded sharp Cheddar
1 cup chicken stock
1/2 cup dry white wine
1/2 teaspoon paprika
Preheat the oven to 350 degrees F and place a rack in the top third of the oven. Grease a 9-by-13-inch baking dish with some olive oil and set aside.
In a medium bowl add the breadcrumbs and the olive oil. Season with salt and pepper and set aside.
Place the chicken in a large pot and cover with water. Add 4 teaspoons salt and bring to a boil, then immediately reduce to a low simmer. Cook the chicken until the internal temperature on an instant read thermometer reads 160 degrees F, 15 to 20 minutes.
Remove and cool slightly, then slice the chicken into thin strips or shred, and place in the bottom of the prepared dish. Add the broccoli and sprinkle the Parmesan over the top.
Melt the butter in a medium saucepan set over medium heat. Add the flour and use a wooden spoon or whisk to stir until smooth and cooked, 4 to 5 minutes. Add the nutmeg, and cayenne. Slowly whisk in the milk, Cheddar, stock and wine. Bring to a boil, whisking often. Reduce to a simmer and cook until thickened and smooth, 5 to 8 minutes.
Pour the sauce over the broccoli and chicken and then top with the breadcrumb mixture and sprinkle the paprika over the top. Cover the dish with foil and bake 15 minutes. Remove the foil and bake until the top is golden and the casserole is bubbling, 15 minutes.
Cool for about 5 minutes and then serve.