Preheat the oven. Set your oven to 375°F (190°C).
Lightly grease a 9x13-inch baking dish or similar casserole pan.
Sauté the aromatics. Warm olive oil in a large skillet over medium heat. Add onion, carrots, and celery with a pinch of salt. Cook 5–6 minutes until softened, then stir in garlic for 30 seconds.
Brown the turkey. Add the ground turkey to the skillet.
Break it up and cook until no longer pink, about 5–7 minutes. Sprinkle in thyme, paprika, oregano, pepper, and remaining salt. If the pan is dry, add a splash of broth to loosen browned bits.
Combine the base. In a large bowl, add uncooked rice, the turkey-vegetable mixture, frozen peas, and chicken broth.
Stir in Greek yogurt (or sour cream) and half of the cheese. Taste and adjust seasoning—this is your best chance to balance salt and spices.
Transfer to the baking dish. Spread the mixture evenly in the prepared pan. Cover tightly with foil to trap steam for the rice.
Bake until rice is tender. For white long-grain rice, bake 30–35 minutes covered.
For brown rice, plan on 50–60 minutes and add 1/2 cup more broth. Check and add a bit more broth if it looks too dry before rice is fully cooked.
Add the topping. Remove foil, fluff gently, and sprinkle on the remaining cheese. If using breadcrumbs, mix them with melted butter and scatter over the top.
Bake uncovered another 8–10 minutes until cheese melts and the top turns lightly golden.
Rest and garnish. Let the casserole sit 5–10 minutes so it sets and slices cleanly. Finish with chopped parsley.
Serve. Scoop into bowls or slice into squares. It’s great with a simple green salad or roasted broccoli.