Preheat the oven to 350°F (180°C) or 160°C if using a fan oven, 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. Stir until all the ingredients are well combined.
Open the cans of crescent rolls 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.
Place a spoonful of the chicken and broccoli mixture on the wider ends of the triangle forming a circle. Roll up the dough from the narrow end, enclosing the mixture.
Lightly brush the crescent ring with the beaten egg. This will help turn it shiny and golden brown as it bakes.
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 for extra flavor and a more appealing look. Slice and serve the rolls warm.
