Preheat the oven to 375 degrees.

Melt the 4 tablespoons of butter in a large skillet or oven-safe pan over medium heat. Add the onion and carrots and cook for 3 to 5 minutes, until soft. Add the mushrooms and garlic and cook for 5 more minutes, until the mushrooms are soft and releasing liquid. Add the thyme, 1/2 teaspoon salt, and 1/4 teaspoon black pepper and cook for 1 minute, until the thyme is fragrant.

Add the flour and stir to coat the vegetables.

Add the chicken broth and cream and bring to a simmer. Simmer for 2 to 3 minutes, until the liquid reaches a gravy-like consistency.

Fold in the chicken, frozen peas, and 2 tablespoons of the parsley. Remove the pan from the heat. Season to taste with salt and black pepper.

If you’re not using an oven-safe pan, transfer the chicken mixture to a baking dish (I prefer shallow baking dishes or pie plates so you can enjoy more of the phyllo topping).

Arrange the phyllo sheets on a flat surface. Working with one sheet at a time, brush the phyllo with a thin layer of the melted butter (keep remaining phyllo sheets covered with a kitchen towel so they don’t dry out). Using your hands, gently scrunch up the phyllo sheet from the middle and nestle it on top of the chicken mixture. Continue until the entire top is covered in phyllo. If necessary, use some of the leftover phyllo to fill in any gaps.

Bake for 10 to 15 minutes, until the phyllo is golden brown, and the filling is bubbling.

Top with the remaining parsley and serve.