3 tablespoons unsalted butter
2 large carrots, halved and thinly sliced
2 ribs celery, diced
1/2 cup yellow onion, chopped
2 cloves garlic, minced
1 teaspoon dried thyme
1/3 cup all-purpose flour, plus more as needed
2 cups chicken broth
1 cup heavy cream
3 cups cooked, shredded chicken
1 cup frozen peas
4 sheets puff pastry, thawed
Kosher salt and freshly ground black pepper, to taste
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!