In a large skillet, melt the butter over medium-high heat. Add the onion, carrots, celery, and garlic. Cook for 5 minutes or until the carrots start to soften, stirring frequently.

Stir in the flour. Cook, stirring constantly, for 2 minutes.

While stirring, slowly pour in the warm broth and heavy cream. Stir in the thyme, salt, and pepper. Reduce the heat to low and bring to a simmer. Cook just until the sauce is thickened, about 5 minutes.

Stir in the chicken and peas. Remove from the heat and transfer the filling into a bowl to cool for 20 minutes.

Meanwhile, preheat the oven to 425°F.

On a lightly floured surface, roll one pie crust to a 12-inch circle. Transfer to a deep 9-inch pie pan, pressing into the pan and up the sides, letting the excess hang over the edge. Add the cooled filling and spread it into an even layer.

Roll out the remaining pie crust into an 11-inch circle and place it on top of the filling. Trim the excess of both pie crusts within 1 inch of the edge of the pie plate. Tuck the excess under itself and crimp the edges as desired. Brush the pie crust with heavy cream. Cut a few slits in the top of the crust to vent steam. Place the pot pie on a rimmed baking sheet.

Bake for 20 minutes. Reduce the oven temperature to 350°F and continue baking until the crust is a deep golden brown and the filling is bubbling, about 40 minutes.

Let the pie cool for 20 minutes before serving. Sprinkle with flaked salt and pepper and garnish with thyme, if desired, before slicing.