Preheat the oven to 400°F/200°C.
In a large pot over medium heat, melt the butter, then add the onion and carrot. Cook, stirring frequently, until the veggies start to soften, about 4-5 minutes.
Add in the mushrooms, frozen peas, frozen corn, garlic, thyme, salt, and pepper. Cook, stirring frequently, until the mushrooms are cooked, about 5-6 minutes.
Sprinkle the flour on the vegetable mixture, then slowly pour in the broth, mixing well to avoid any clumps.
Add the milk and the shredded chicken and allow the filling to lightly simmer until the sauce has thickened enough that when you run a spatula across the bottom of the pan, the sauce doesn't fill in the space right away. This should take 8 to 10 minutes.
While the sauce is thickening, roll out both pie crusts. The bottom crust should be roughly 13 inches in diameter, and the top roughly 10 inches.
Place the bottom pie crust into a 9 inch pie pan, pour the filling into the crust, then top with the remaining pie crust. Press to seal the top and bottom crust together on the edges. Slice a few vent holes in the top.
Brush the top of your chicken pot pie with egg wash. Transfer the pie to the oven and bake for 30 minutes, or until the filling is bubbly and the top is golden brown.
Sprinkle with fresh tyme if desired. Serve warm.