Preheat oven to 400°F.
Heat 1 tbsp olive oil or butter in a large skillet over medium heat.
Add 1/4 small onion, chopped, and sauté 3–5 minutes until translucent.
Add 4 oz mushrooms, sliced, and sauté 2–3 minutes until softened.
Add 3/4 cup chopped broccoli and 1 cup cooked chicken, cubed; cook 2 more minutes until broccoli is bright and chicken is heated through. Remove from heat and set the filling aside to cool slightly.
Line a large baking sheet with parchment paper.
Carefully unfold both thawed puff pastry sheets on a clean work surface. Cut each sheet into two equal squares so you have 4 pieces total. Gently roll each square slightly to form a rectangle.
Divide the cooked filling evenly among the 4 puff pastry rectangles, placing the filling in the center of each and leaving a small border around the edges.
Top the filling on each rectangle with the shredded cheese, dividing 1 cup shredded cheese evenly among the 4 pieces.
Fold the pastry sides over the filling to enclose (forming a packet) and pinch the edges firmly to seal.
Place the sealed puffs seam-side down on the prepared baking sheet, leaving space between them.
Bake for 20 minutes or until the puffs are golden brown and puffed.
Cool 2–3 minutes before serving.