Preheat your oven to 400°F (200°C) and set out your pie crusts to come to room temperature for easier rolling.
In a large skillet, melt the butter over medium heat. Add the diced onion and minced garlic, cooking until the onion becomes translucent and fragrant, about 3-4 minutes.
Add the sliced mushrooms to the pan. Sauté until they release their moisture and start to brown, about 5-7 minutes.
Stir in the flour to create a light roux, cooking for about 1 minute to remove any raw flour taste.
Slowly whisk in the chicken broth, then the milk and heavy cream. Bring the mixture to a simmer; it will thicken into a creamy sauce.
Season with thyme, salt, and pepper. Stir in the shredded chicken and frozen peas, allowing everything to heat through and the flavors to meld, about 4-5 minutes.
Roll out the bottom crust and fit it into a pie dish. Spoon the creamy filling into the crust, spreading evenly.
Lay the second crust over the filling. Trim excess, seal the edges, and cut a few vents on top to let steam escape.
Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown and the filling is bubbling around the edges.
If the crust browns too quickly, cover the edges with a foil shield and continue baking until done.
Allow the pie to rest for about 10 minutes before serving so the filling can firm up for clean slices.