In a 12-inch cast-iron (or oven-safe) skillet set over medium-low heat, add the onion, garlic, sage, thyme butter and 1/2 teaspoon kosher salt. Cook, stirring occasionally, until the onions are translucent, about 5 minutes.
Sprinkle the flour over the onion mixture and cook, stirring constantly, for 2 minutes.
Stir in the chicken broth and 1 cup heavy cream, scraping up any brown bits from the bottom of the pot. Bring the mixture to a boil then reduce it to a simmer and cook until the mixture has thickened slightly, about 3 minutes.
Stir in the chicken, and peas and carrots, then taste and season the mixture with salt and pepper. Remove the mixture from the heat and let it cool completely before covering it with plastic wrap and storing it in the fridge.
When ready to bake, preheat the oven to 425°F.
Unfold the puff pastry on your work surface and cut it into 2-inch squares, rectangles, triangles or circles. Arrange the puff pastry pieces on the top of the pot pie mixture so that they are overlapping slightly.
In a small bowl, whisk the egg with the remaining 1 tablespoon heavy cream. Brush it generously atop the puff pastry, then bake the pot pie until the pastry is golden brown and puffed, about 25 minutes.
Remove the pot pie from the oven and let it cool for 15 minutes before serving.