Adjust an oven rack to a low-center position and preheat to 400 degrees F (200 degrees C). Press one pie pastry into each of two 9-inch deep-dish pie pans. Reserve remaining pastries until needed.

Combine broth and evaporated milk in a microwave-safe bowl; microwave on high until steamy, 3 to 4 minutes.

At the same time, heat oil in a Dutch oven over medium-high heat. Add onions and celery; sauté until just tender, about 5 minutes. Transfer to a bowl.

Melt butter in the Dutch oven over medium heat. Whisk in flour and thyme, and cook until golden, about 1 minute. Whisk in hot milk mixture and bring to a simmer. Continue to simmer, stirring occasionally, until sauce thickens, about 1 minute.

Remove the Dutch oven from the heat. Stir in wine, salt, and pepper. Add onion mixture, shredded chicken, frozen vegetables, and parsley; stir until well combined.

Divide chicken mixture between the prepared pie pans. Place remaining pastries over top, and flute the edges by pinching with your fingers. Place both pie pans onto a baking sheet.

Bake in the preheated oven until pastries are golden brown and filling is bubbly, 30 to 35 minutes.