Preheat oven to 400°. Sprinkle chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper.
Heat 1 tablespoon oil in a large, deep skillet over medium-high heat. Add chicken and cook, stirring often, 8 minutes or until well browned. Transfer chicken to a plate.
Add remaining 1 tablespoon oil to skillet. Add turnip and next 3 ingredients; cook, stirring often, 10 minutes or until softened. Add flour and garlic and cook, stirring constantly, 1 minute. Gradually whisk in broth and half-and-half and cook, whisking constantly, until mixture comes to a boil. Reduce heat to low, and simmer.
Stir in chicken, remaining 1/2 teaspoon salt and remaining 1/4 teaspoon pepper. Stir in Swiss chard and thyme, and simmer 5 more minutes or until vegetables are tender. Remove from heat, and stir in lemon juice.
Spoon vegetable mixture into a lightly greased 9-inch square baking dish. Place puff pastry over filling, tucking edges of pastry into dish, if necessary. Cut several slits in pastry to vent steam; brush top of pastry with egg.
Place baking dish on a baking sheet. Bake 20 minutes or until pastry is golden.
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