Preheat the oven to 400°F. Lightly oil a 9-inch pie dish and set aside.

Heat oil and butter in a large skillet over medium high heat.

Add chicken and cook, stirring occasionally, until no longer pink, 4 to 5 minutes.

Add mushrooms, 1 teaspoon of the thyme, carrots, celery, onions, 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper and cook until just tender, 6 to 8 minutes.

Add flour and cook, stirring constantly, for 1 minute. Add broth and milk, stir well and bring to a simmer.

Reduce heat to medium low and simmer until very thick, about 10 minutes.

Meanwhile, combine rice, cheese, paprika, remaining 1 teaspoon thyme, remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper in a medium bowl.

Stir peas into chicken mixture and transfer to prepared dish.

Scatter rice mixture evenly over the top then arrange dish on a baking sheet to catch any drips.

Bake until bubbly and crust is crisp, about 30 minutes.