In a large mixing bowl, combine warm milk, sugar, and active dry yeast. Stir gently and let sit for 5-7 minutes until foamy.
Add melted butter, egg, and salt to the yeast mixture and whisk to combine.
Gradually add the flour, mixing until a soft dough forms. Knead the dough on a lightly floured surface for about 8-10 minutes until smooth and elastic.
Place the dough in a lightly greased bowl, cover with a clean towel, and let it rise in a warm place for 1 hour or until doubled in size.
While the dough is rising, prepare the filling. In a medium bowl, mix shredded chicken, chopped broccoli, shredded cheddar cheese, cream cheese, sour cream, mayonnaise, garlic powder, onion powder, salt, and black pepper until well combined.
Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper.
Punch down the risen dough and divide it into 8 equal portions.
Roll each portion into a 6-inch circle on a lightly floured surface.
Place about 3 tablespoons of the chicken broccoli cheese filling in the center of each dough circle.
Carefully fold the edges of the dough over the filling, pinching to seal and form a bun. Place each bun seam-side down on the prepared baking sheet.
Cover the buns with a clean towel and let them rest for 15 minutes to rise slightly.
Bake the buns in the preheated oven for 18-20 minutes or until golden brown on top.
Remove from the oven and let cool for 5 minutes before serving warm.