Preheat the oven to 425 degrees. Line a baking sheet with parchment paper and mist with cooking spray.
On a lightly floured surface, roll the puff pastry into a 14-inch square and cut into four 7-inch squares. Brush the center of each with mustard and season with salt and pepper. Place the mushrooms gill-side up on the pastry and season with salt and pepper.
Toss the breadcrumbs and cheese in a bowl and sprinkle over the mushrooms. Fold in the pastry edges, pleating to fully cover the mushrooms; press to seal. Brush both sides of the pastry with the beaten egg. Place seam-side down on the baking sheet, make slits in the top and bake until golden and crisp, 25 minutes. Let cool slightly.
Meanwhile, combine the shallot, vinegar, and salt and pepper to taste in a bowl, then whisk in the olive oil. Toss the dressing with the salad greens and pear. Serve the salad with the mushroom pies.