Preheat your oven to 400 degrees and remove the puff pastry from freezer or fridge to thaw.
In a large frying pan, melt the butter over medium heat. Add the chopped onion and let it saute and soften for a couple of minutes.
Add the celery and the minced garlic. Stir and let saute for a couple of minutes.
Add the potatoes and carrots, and stir. Let it saute for a couple of minutes.
Sprinkle the tops of the veggies with the flour. Continuing stirring the combine the flour with the veggies for 1 minute.
Add the chicken broth and stir until combined. Let the chicken broth start to heat for a couple of minutes.
Add the milk. Stir to combined. The mixture with is thickening by now. Let it simmer for a couple minutes, stirring occasionally.
Add the thyme, salt, and pepper and stir it well to combine. Add the cooked chicken and stir well. Turn the heat to low and let it simmer about 5 minutes as you prep your crust.