In a medium bowl, combine the warm milk and sugar. Sprinkle the yeast over the milk and let it sit for 5-7 minutes until foamy.
In a large mixing bowl, combine the flour and 1 teaspoon salt. Add the yeast mixture, melted butter, and egg. Mix until a dough forms.
Turn the dough onto a floured surface and knead 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 large bowl, mix the 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.
Once the dough has risen, punch it down and divide it into 8 equal pieces.
Roll each piece into a circle about 6 inches in diameter on a lightly floured surface.
Place about 1/4 cup of the chicken and broccoli filling in the center of each dough circle.
Carefully fold the edges of the dough over the filling and pinch to seal, forming a bun. Place the buns seam side down on the prepared baking sheet.
Cover the buns loosely with a towel and let them rest for 15 minutes to rise slightly.
Bake the buns in the preheated oven for 20-25 minutes, or until golden brown on top.
Remove from the oven and let cool slightly before serving.