Heat oven to 375°F Oil four 12 oz ramekins, or use a 9" pie plate.
Heat the oil in a large skillet on medium heat. Add the onion, salt and pepper. Cook, covered, stirring occasionally, for 6 minutes. Add the sweet potatoes and cook, covered, stirring occasionally until tender, 8 to 10 minutes.
Meanwhile, shred chicken discarding bones and skin; set aside.
Sprinkle flour over the vegetables and cook, stirring, for 1 minute. Gradually stir in the wine and then the broth and bring to a boil.
Add the chicken, peas, parsley, tarragon, and nutmeg.
Divide the mixture among the prepared ramekins (about 3/4 C each).
Using a 4" round cookie cutter, cut out 4 pieces of puff pastry. Place the pastry on top of the ramekins.
Place the pot pies on a rimmed baking sheet (line with foil to catch drips). Brush pastry with beaten egg. Bake until puffed and golden brown, 20 to 25 minutes.