Unfold the thawed puff pastry and seal seams using a rolling pin. Dock the sheet by piercing it several times with a fork. Cut into 12 (2-inch) rounds, place on a half sheet pan, and refrigerate while preparing the rest of the casserole.
Heat the oven to 400ºF.
Melt 1 tablespoon of the butter in a 12-inch straight-sided saute pan over medium heat. Add the onion and celery and sweat until the onion is translucent, about 5 minutes. Add the frozen vegetables and cook, stirring occasionally, until heated through, 7 to 8 minutes.
While the vegetables are cooking, combine the broth and milk in a 2-cup microwave-safe container and heat in the microwave until almost boiling, about 2 minutes.
Add the remaining 2 tablespoons butter to the celery mix and melt. Add the flour and curry powder and whisk for 1 to 2 minutes. Whisk in the hot milk mixture and cook until thickened, 3 to 4 minutes. Add the parsley, salt, and pepper. Add the chicken, and stir to combine. Pour into a 13-by-9-inch baking dish and mash down the mixture to compact the casserole. Top with the circles of puff pastry, spacing them so that they do not touch. Bake on the middle rack of the oven until the puff pastry has browned and the mixture is hot and bubbly, about 25 minutes. Cool for 30 minutes before serving.