Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the diced onion and cook until softened, about 3-4 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Add the chicken pieces to the skillet and cook until they are no longer pink on the outside, about 5 minutes.
Add the diced carrots, celery, and sliced mushrooms to the skillet. Cook for another 5 minutes until the vegetables begin to soften.
Stir in the thawed peas, dried oregano, dried thyme, salt, and black pepper. Mix well.
Sprinkle the flour over the mixture and stir to coat evenly. Cook for 1 minute to remove the raw flour taste.
Pour in the chicken broth and bring the mixture to a simmer. Cook for 5 minutes until the sauce thickens slightly.
Remove the skillet from heat and stir in the Greek yogurt until fully incorporated and creamy.
Transfer the chicken and vegetable mixture to a 9×13 inch baking dish. Sprinkle the shredded mozzarella and grated Parmesan evenly over the top.
In a small bowl, combine the panko breadcrumbs with the melted butter and chopped parsley. Sprinkle this mixture evenly over the cheese layer.
Bake uncovered in the preheated oven for 20-25 minutes, or until the topping is golden brown and the casserole is bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.