Melt the butter in a large skillet over medium-high heat. Add the onion, carrots, and celery. Reduce heat to medium-low and cook for 3-5 minutes until softening.
Add the flour and stir to combine. Cook for another minute or two.
Add 3 cups of the chicken stock (and wine, if using) and stir to combine. Bring back to bubbling and let thicken for 3-5 minutes.
Add the chicken, turmeric, thyme, and red pepper flakes (if using). Stir to combine. Season with salt and pepper.
Add the heavy cream. Stir to combine, bring back to bubbling, and let thicken for 3-5 minutes. If too thick, add a little more stock.
Remove from heat and let cool to room temperature.
Roll out the pie crusts and cut out bottom (~7.5" diameter) and top (~5" diameter) circles. Spray the pans with cooking oil and add the bottom crust. Add enough filling then add the top. Cut off excess and crimp the edges with a fork. (It seems like two sheets isn't enough but with rolling, using excess, re-combining, you'll actually have some left over.)
Mix the egg with a tablespoon of water until frothy. Brush the tops of the pies. Slice vents in the top of the pies.
Heat oven to 350°. Place the pies on a foil-lined baking sheet -- if any filling 'cooks out and over' you'll thank me for this -- and bake 30-35 minutes until the crust is a golden brown.
Serve straight out of the pans or turn them out and serve upside down. Or freeze (see notes).¼