Preheat oven to 400°F and grease a 13×18-inch sheet pan with nonstick spray. Set aside.
In a large saucepan over medium-high heat, melt the butter. Add carrot, celery, onion, garlic, and thyme, and season with salt and pepper. Cook until the vegetables are tender but still crisp, about 5 minutes.
Stir in the flour and let cook 1 minute. Add the broth and cook until starting to thicken, about 5 minutes. Stir in the cream and continue cooking until sauce has thickened, about 2 minutes.
Add cooked chicken and peas and season to taste with salt and pepper. Pour mixture into prepared sheet pan. Set aside.
Lightly dust a surface with flour, then stack 2 sheets of puff pastry atop the surface. Roll out into a 15-inch square, then cut the pastry into 1-inch strips. Repeat with remaining sheets of puff pastry. (You should have 30 strips total.)
Arrange puff pastry strips over chicken filling in a row, overlapping them a bit so most of the filling is covered. Trim overhang as needed.
In a small bowl, beat the egg with a bit of water and brush pastry with egg wash.
Bake until pastry is a deep golden brown, 35-40 minutes. Enjoy!
