Make the stew: Season pork with 2 teaspoons salt and 1/2 teaspoon pepper. Heat a large, heavy pot over high heat for 1 minute. Add 2 tablespoons oil. Working in batches, brown pork on all sides, 3 to 4 minutes total per batch. Using a slotted spoon, transfer pork to a large bowl. (Reduce heat slightly if bottom of pot is getting too dark; you want browned, not burned, bits for flavoring the stew. Either wipe out pot or deglaze with water, bringing it to a boil and scraping up burned bits. Pour off liquid, add more oil, and proceed.)
Add carrot, celery, and onion to drippings in pot. Add cup wine, and stir, scraping up any browned bits from bottom, until wine is evaporated. Cook for 3 minutes more, then add remaining 2 cups wine. Cook, stirring often, until reduced by half, 6 to 8 minutes. Whisk caraway seeds, mustard, flour, and 1/2 cup stock in a small bowl, then whisk into pot. Bring to a boil, and cook, stirring often, for 30 seconds.
Stir in remaining 3 1/2 cups stock. Add pork, thyme sprigs, bay leaf, and 1/2 teaspoon salt, and bring to a boil. Reduce heat to a simmer, and cook, partially covered, until pork is fork-tender, about 1 hour and 30 minutes.
Finish the pies: Preheat oven to 375 degrees. On a lightly floured surface, working with 1 sheet of puff pastry at a time, roll out each to 1/8 inch thickness. Using a 3 1/2-inch fluted round cutter, cut out 12 circles, and arrange on a parchment-lined baking sheet. Cut out shapes from center of each round using a 1-inch snowflake cutter if desired. (Alternatively, make a wreath: Using a 1 1/2-inch leaf cutter, cut leaves from dough. Brush with egg, and form into wreath shapes about 3 1/2 inches in diameter, using 10 leaves per wreath. Save 1 leaf to place in center.) Brush with egg, then freeze until firm, about 15 minutes. Bake puff pastry rounds until golden brown, about 15 minutes.
Meanwhile, add remaining vegetables to pork mixture, and bring to a boil. Reduce heat to a simmer, and cook until vegetables are just tender, 15 to 18 minutes. Season with salt and pepper, and stir in parsley and thyme. Divide among twelve 7-ounce ramekins. Place on a rimmed baking sheet.
Place 1 puff pastry round on each ramekin. Bake, rotating sheet halfway through, until pastry is golden brown and juices are bubbling, 20 to 25 minutes more. Let stand for 15 minutes before serving.