Heat 1 tablespoon oil in a 10-inch cast-iron skillet (or other ovenproof skillet) over medium-high heat.  Sprinkle chicken with 1/2 teaspoon salt and 1/2 teaspoon pepper.  Add chicken to pan; cook 6 minutes on each side.  Place chicken on a cutting board; let stand 10 minutes.  Shred into large pieces.

Preheat oven to 425°F.

Combine 2 1/2 cups chicken stock and flour in a bowl, stirring with a whisk.  Add remaining 1 tablespoon oil to the pan.  Add onion, carrot, garlic, and thyme; sauté 6 minutes.  Add remaining 1/2 cup stock, scraping pan to loosen browned bits.  Stir in kale or turnip greens; cook 1 minute.  Stir in remaining 1/2 teaspoon salt, remaining 1/2 teaspoon pepper, and flour mixture.  Add turnips and bring to a boil.  Reduce heat, and simmer for 15 to 25 minutes or until vegetables are tender.  Check for seasoning; add salt and pepper if needed.  Stir in chicken and parsley.  Remove pan from heat.

Arrange pie crust over the skillet, folding edges under as needed.  Cut slits in dough to allow steam to escape.  Brush with egg white.  Bake at 425°F for 20 minutes.  Cool for 10 minutes before serving.