Preheat the oven to 350°F (180°C), and line a large baking sheet with parchment paper. In a bowl, mix together the shredded chicken, chopped broccoli, softened cream cheese, mozzarella, cheddar cheese, garlic, salt and pepper.
Open the cans of crescent dough and unroll, separating the triangles and place on the prepared baking sheet. Overlap the wide sides of the crescent roll triangles to form a ring. It will look like a sun.
Bake in the preheated oven for 18-20 minutes, or until the crescent ring is golden brown and cooked through. If desired, sprinkle a bit of dried parsley or Italian seasoning on top of the rolls. Slice and serve the rolls warm.