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In a large skillet, melt 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.
Stir in the chicken and peas. Pour the filling into a bowl to cool.
Preheat the oven to 425F.
On a lightly floured surface, roll one pie crust to a 12-inch circle. Transfer to a deep 9-inch pie plate. Pour in the cooled filling.
Roll out the remaining pie crust and place it on top of the filling. Trim the excess crust within 1 inch of the edge of the pie plate. Tuck the edges under and crimp as desired. Brush the pie crust with heavy cream. Cut a few slits in the top of the crust to vent steam. Place on a rimmed baking sheet.
Bake for 20 minutes.
Reduce the heat to 350F 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.