In a large skillet, heat butter over medium-high heat; saute mushrooms and shallots until tender, 3-4 minutes. Add garlic; cook and stir 1 minute longer. Stir in flour until blended. Gradually stir in broth and wine; bring to a boil, stirring constantly. Reduce heat; simmer, uncovered, until slightly thickened, 2-3 minutes, stirring occasionally. Remove to a bowl; cool slightly. Refrigerate, covered, until cold., Preheat oven to 425°. Pound chicken breasts with a meat mallet to even thickness. Whisk together egg and water. On a lightly floured surface, roll pastry sheet into a 14-in. square. Cut into four 7-in. squares. Place 1 chicken breast on center of each square. If desired, fold narrow end under to fit pastry square; sprinkle with salt and pepper. Top each with 1 rounded tablespoon mushroom mixture. Lightly brush pastry edges with egg mixture. Fold pastry over filling; press edges with a fork to seal. Place on a rimmed parchment-lined baking sheet., Brush tops with egg mixture. Bake until golden brown and a thermometer inserted in chicken reads 165°, 18-23 minutes. Reheat remaining mushroom mixture; serve with pastries. If desired, garnish with thyme.