Preheat the oven to 400 degrees F. Spray an 18-by-13-inch sheet pan with cooking spray.
Melt the butter with the thyme in small saucepan over medium heat. Whisk in the flour and cook, stirring, until a sandy consistency, about 1 minute. Pour in the chicken broth, whisking constantly until smooth. Stir in 2 teaspoons salt and a few grinds of black pepper and bring to a boil. Lower the heat and simmer gently until thickened and the flour flavor is gone, about 10 minutes.
Meanwhile, add the chicken, onions, carrots and celery to the prepared sheet pan and toss to combine; set aside.
Stack 2 puff pastry sheets on top of each other on a lightly floured surface. Roll the puff pastry out to a 15-by-15-inch square. Cut the pastry into 1-inch strips so you have 15 strips total. Repeat with the remaining 2 puff pastry sheets.
Pour the gravy over top of the chicken mixture on the sheet pan. Arrange the pastry strips over the filling lengthwise, overlapping them slightly so most of the filling is covered. Trim any overhang from the edges of the pan. Brush the pastry with the egg wash and bake until deep golden brown, 35 to 40 minutes.
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