4 boneless, skinless chicken breast halves
Meanwhile, bring a medium pot of salted water to a boil and prepare an ice-water bath. Add baby carrots to boiling water and cook until tender, about 5 minutes. Drain and immediately transfer to ice-water bath until cool. Drain; set aside. Place peas in boiling water and cook about 30 seconds; drain and set aside. Add pearl onions to boiling water and cook about 1 minute, transfer to ice-water bath until cool. Drain; set aside.
Heat 1 teaspoon butter in a medium skillet over medium heat. Add mushrooms and cook until browned, 3 to 4 minutes; let cool.
Divide mixture evenly between four 13-ounce shallow baking dishes. Cut puff pastry into four 8-inch circles and place on top of each of the bowls, crimping edges as desired. Cut a slit in the center of each piece of puff pastry to allow steam to escape; brush beaten egg over puff pastry.
Transfer bowls to a baking sheet. Bake until puff pastry is golden brown and filling is bubbling, about 25 minutes. Let stand 5 minutes before serving.