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Preheat oven to 425 degrees F. Set chicken thighs in a medium baking dish. Drizzle with olive oil and sprinkle generously on both sides with salt and pepper. Roast 20 - 25 minutes, or until internal temperature reaches 165 degrees F. Set aside to cool, then shred or chop chicken.
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Meanwhile, fill a wide, deep-sided pot with water and bring to a boil. Add 2 tablespoons salt. Blanch carrots, beets, rutabaga, and turnips for 3 minutes. Drain and set aside.
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Melt butter in the same pot over medium heat. Add leeks, onion, and thyme sprigs. Cook 10 minutes, stirring often. Stir in flour, along with 1/2 teaspoon each salt and pepper, and cook 2 minutes. Slowly stir in warm broth, allowing mixture to bubble and thicken between additions. Cook until mixture is nice and thick, 20 minutes total. Fish out the thyme sprigs and discard.
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Fold in chicken and any accumulated juices, blanched vegetables, and parsley. Add salt and pepper to taste.
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Spoon mixture into a 3-quart 10-inch skillet, Dutch-oven, or 9 x 13-inch baking dish.
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On a lightly floured surface, roll pastry into a 14-inch square for a round or square baking dish, or an 11 x 15-inch rectangle for a rectangular one. Place on top of the filling. For a round baker, fold the corners up over the crust. I let my crust kind of float on top of the filling, but you do run the risk of it sinking into the filling. For best results, crimp the overhanging crust over the edges of the baking dish. Brush crust with egg wash, and sprinkle with a pinch of salt and pepper. Use a sharp knife to cut 4 (3-inch) slits in the top.
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Slide into oven and turn heat down to 375 degrees F. Bake 40 - 50 minutes, or until filling is bubbling vigorously and pastry top is a deep golden brown.
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Cool 20 minutes, slice, and serve.