Set a rack in the center of your oven and preheat to 350 degrees F. Spray a 9×13 casserole dish with nonstick spray.
Heat the olive oil and butter in a Dutch oven or other large pot with a sturdy bottom over medium heat. Add the onion, salt, and pepper once the butter has melted.
For about 8 to 10 minutes, cook the onion until it is soft and translucent. The onion should soften without browning, so stir occasionally and adjust the heat as necessary. Stir in the garlic and cook for 30 seconds.
Then sprinkle the flour over the top. Stir continuously for at least 1 minute, or until all white bits are gone from the flour.
Pour the milk and broth in slowly, stirring constantly to prevent lumps. If any lumps form, break them up as needed. Stirring frequently, bring the mixture to a simmer, then reduce the heat and let gently bubble until the sauce has thickened, about 5 minutes. Adjust the heat as necessary to maintain a rapid simmer.
Turn off the heat and remove the pan. Stir in the veggies, Greek yogurt, half of the cheddar cheese, mozzarella, Dijon, and thyme until smooth.
Stir in the shredded chicken and cooked rice.
Place the mixture in the prepared casserole dish. Top with the remaining cheese.
Bake your Pioneer Woman Chicken Rice Casserole until the cheese is melted and it is hot about 15 minutes. Turn the oven on to broil, and continue cooking until the cheese is lightly browned about 3 additional minutes. Be careful not to burn the cheese.
Remove from the oven and sprinkle with fresh parsley or thyme. Let rest for 5 minutes, then serve hot.