Preheat oven to 400 degrees F. Grease a 2-quart baking dish with 1 tablespoon butter.

Melt remaining butter in a large skillet over medium heat. Add onion and celery and cook until softened, about 2 minutes. Season with 1 teaspoon salt and 1/2 teaspoon pepper. Add chicken, broccoli, and parsley and combine. In a medium bowl, beat 6 eggs and milk. Set aside.

Spread half the bread cubes on the bottom of the baking dish. Layer chicken mixture evenly over bread cubes and sprinkle with half the cheese, remaining bread, and egg mixture.

Bake until cooked through, about 20 minutes. Sprinkle with remaining cheese and bake until cheese is melted and strata is golden, 5 minutes more. Garnish with parsley and serve immediately.