Preheat the oven to 375 degrees F (190 degrees C). Lightly coat a baking dish with cooking spray.

Melt butter in a large skillet over medium heat. Add celery, onion, and 1 teaspoon salt; cook and stir until onion begins to turn translucent, about 5 minutes. Season with black pepper, sage, thyme, rosemary, and cayenne pepper; cook and stir about 1 minute. Off heat, stir in chicken broth.

Combine crushed crackers and onion mixture in a large bowl. Add milk and egg; stir until combined. Season with salt and black pepper. Transfer to the prepared baking dish; smooth top.

Bake in the preheated oven until top is golden brown, 40 to 45 minutes.