Preheat 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 turkey, mushrooms, 1 teaspoon of the thyme, carrots, celery, onions, salt and 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, salt and pepper in a medium bowl.
Stir peas into turkey mixture then season with salt and pepper 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.