Preheat the oven to 425ºF. Over medium heat, heat the butter in a Dutch oven, large pot or large skillet (12-inch diameter ) and let melt completely. Add the root vegetables, onion, celery and garlic and stir with 1 teaspoon salt (Diamond Crystal kosher, but adjust according to your salt’s saltiness). Cook, stirring regularly, until the root vegetables are soft on the outside but still firm in the middle, about 15 minutes.
Add the chicken to the Dutch oven, stirring it in with the vegetables. Sprinkle the flour over the chicken and vegetables. Toss and stir to combine very well to coat each piece. Cook, stirring constantly, for about 2 to 3 minutes, then pour in the milk, creating a thick, yet stew-like consistency. Pour in the broth, stir again and once hot and bubbly, add the spinach, capers and thyme. Stir until the leaves are mostly wilted and most of the chicken pieces aren’t pink anymore. Season to taste with salt and pepper, stir again then turn off the heat.
Pour the chicken and vegetable mixture into a 8 x 8 x 2-inch square or round pan and level it out. Then on a cutting board, cut a piece of puff pastry that’s just slightly bigger than you need to cover the pan, so there’s about a ½-inch overhang. Press it over the pan and filling, and make a few slits in the pastry.
Brush the pastry with the egg, place the pan on a sheet pan and place in the oven on the middle rack. Bake until the pastry has puffed, turned golden to golden brown and is bubbling over a bit onto the sheet pan, about 30 to 35 minutes. Let cool on a rack for 10 minutes before serving. Serve the pot pie hot.