Heat olive oil and butter in a large skillet over medium heat. Season chicken breasts with salt and black pepper. Cook chicken in the hot oil until golden and almost cooked through, about 4 minutes per side. Transfer to a plate.

Place onion, mushrooms, and garlic in the same skillet; cook and stir until mushrooms lose their moisture and onions are tender, about 5 minutes. Stir in parsley.

Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.

Roll out each puff pastry sheet into a 14-inch square on a floured work surface. Cut each sheet into 4 equal squares. Place a chicken breast in the center of each square.

Combine cream cheese and Dijon mustard in a small bowl. Spread some of the cream cheese mixture over each chicken breast. Top each with 2 tablespoons of the mushroom mixture. Brush the edges of each square with water and wrap around the chicken. Place seam-side down on a baking sheet. Brush the tops with the beaten egg.

Bake in the preheated oven until puffed and golden brown, about 20 minutes. Remove from oven and slice in half before serving.