Preheat and prep: Heat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
If using puff pastry, let it thaw according to package directions.
Blanch the broccoli: Bring a small pot of salted water to a boil. Add broccoli and cook for 1–2 minutes until bright green and slightly tender. Drain and rinse under cold water to stop cooking.
Pat dry.
Sauté aromatics (optional but tasty): Warm olive oil or butter in a skillet over medium heat. Add minced garlic and cook for 30–60 seconds until fragrant. Remove from heat.
Make the filling: In a large bowl, combine cream cheese, Greek yogurt or sour cream, sautéed garlic, onion powder, Italian seasoning, Dijon, salt, and pepper.
Stir until smooth. Fold in the shredded chicken, broccoli, and shredded cheese. Taste and adjust seasoning.
Roll the dough: On a lightly floured surface, roll the dough into a large rectangle about 1/8–1/6 inch thick.
Cut into 6–8 equal rectangles or squares. Aim for pieces roughly 4x6 inches.
Fill the pockets: Spoon 1/4–1/3 cup of filling onto one half of each rectangle, leaving a small border. Don’t overfill.
Seal: Brush the edges with a little water.
Fold the dough over the filling to make a pocket. Press edges firmly, then crimp with a fork. Cut 2–3 small slits on top of each pocket to vent steam.
Egg wash: Transfer pockets to the baking sheet.
Brush tops with the egg wash for a glossy, golden finish. Sprinkle a pinch of Italian seasoning or flaky salt if you like.
Bake: Bake for 16–22 minutes, depending on dough type and size, until deep golden and crisp. Puff pastry may cook a bit faster; pie crust may take a touch longer.
Rest and serve: Let the pockets rest for 5 minutes before serving.
The filling will set slightly and be easier to eat.