Preheat your oven to 375°F (190°C). Line a large, rimmed baking sheet with parchment paper or spray lightly with non-stick cooking spray.
In a medium mixing bowl, vigorously stir together the softened cream cheese, mayonnaise, garlic powder, onion powder, salt, and black pepper until completely smooth.
Gently fold the shredded chicken, thoroughly drained chopped broccoli, and 1 cup of shredded cheese into the cream cheese mixture until evenly distributed.
Unroll the crescent roll dough and separate it along the perforated lines into 8 triangles. Place them flat on your clean work surface.
Place roughly 1/3 cup of the filling mixture onto the widest end of each dough triangle.
Fold the outer corners of the wide end slightly over the filling, then roll the pastry forward toward the narrow point. Secure the point underneath the roll.
Arrange the rolled pastries evenly on the prepared baking sheet, leaving about 2 inches of space between each roll.
In a small bowl, whisk the egg and water together. Lightly brush the tops and sides of each roll with the egg wash using a pastry brush.
Sprinkle the remaining 2 tablespoons of shredded cheese directly over the tops of the brushed rolls.
Bake for 15 to 20 minutes, or until the pastry is puffed and a deep golden brown, and the cheese on top is melted and bubbly.
Let the rolls cool on the baking sheet for 5 minutes before transferring to a platter to serve.