Preheat your oven to 400°F.

In a large skillet or Dutch oven, melt the butter over medium heat. Add the diced onions and cook until they are translucent.

Add the sliced mushrooms to the skillet. Season with salt and black pepper, cooking until the mushrooms release their liquid and begin to brown, about 10 minutes.

Sprinkle the flour over the mushroom mixture and then pour in the chicken broth. Stir well to eliminate lumps and form a thick gravy.

Stir in the sugar snap peas and mixed frozen vegetables until evenly mixed.

Season the filling with Kosher salt and black pepper to taste; stir in heavy cream and adjust seasonings as needed.

Add the shredded rotisserie chicken, mixing it in with the vegetables. Fold in fresh thyme leaves and transfer the filling to a casserole dish.

Beat the egg with water to create an egg wash and set aside.

Unwrap and lay the pie dough flat. Slice it into 3/4-inch strips and weave them into a lattice design on parchment paper.

Carefully flip the lattice design over the casserole using the parchment paper. Remove the parchment and adjust the lattice, letting the edges drape over the sides of the dish.

Place the casserole on a rimmed baking tray and bake for 30 minutes, or until the filling bubbles and the crust turns golden brown. Cover with foil if the crust browns too quickly.

Remove the pie from the oven, sprinkle fresh thyme on top, and let it rest for 10 minutes before serving.