Preheat oven to 400°F. In a saucepan over medium heat, melt butter. Add flour; cook, stirring constantly, 1-2 min. Slowly add broth, whisking until smooth; bring to a boil.
Add thyme, bay leaf, onions, celery, carrots and mushrooms. Cook until tender, about 5 min. Add potatoes, chicken, peas, salt and pepper. Cook until potatoes are tender, about 5 min. Remove from heat; cool 5 min. Discard bay leaf.
Divide mixture among 4 ovenproof bowls; place on a baking sheet. Brush pastry with egg mixture. Brush rims of bowls with water. Place 1 pastry square on top of each bowl, pressing edges lightly.
Bake until puffed and golden brown, about 20 min. Let rest 5 min. before serving.
