Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper.
In a large mixing bowl, combine the shredded cooked chicken, finely chopped broccoli, shredded cheddar cheese, softened cream cheese, sour cream, grated Parmesan cheese, chopped onion, garlic powder, salt, and black pepper. Mix well until all ingredients are evenly incorporated.
Open the refrigerated biscuit dough and separate the 8 biscuits. On a clean surface, flatten each biscuit into a 4-inch circle using your hands or a rolling pin.
Divide the chicken and broccoli mixture evenly among the flattened biscuit circles, placing the filling in the center of each.
Carefully fold the edges of each biscuit dough circle up and around the filling, pinching the dough together at the top to seal and form a bun. Make sure the filling is completely enclosed.
Place the sealed buns seam-side down on the prepared baking sheet, spacing them about 2 inches apart.
Brush the tops of the buns lightly with olive oil to help them brown nicely.
Bake in the preheated oven for 18 to 20 minutes, or until the buns are golden brown and cooked through.
Remove from the oven and let the buns cool for 5 minutes before serving.
Garnish with chopped fresh parsley for a pop of color and freshness. Serve warm.