Place a rack in the center of your oven and preheat the oven to 350 degrees F. Coat a 9x13 or similar casserole dish with nonstick spray.
In a Dutch oven or similar large, sturdy-bottomed pot, heat the olive oil and butter over medium heat. Once the butter melts, add the onion, salt, and pepper. Cook until the onion is softened and translucent, about 8 to 10 minutes. Stir occasionally and adjust the heat as needed so that the onion softens but does not brown. Stir in the garlic and let cook 30 seconds.
Sprinkle the flour over the top. Cook and stir constantly for at least 1 full minute, until the flour is turning brown and no white bits remain.
Slowly pour in the broth and milk, stirring constantly to prevent any lumps. Pause as needed to break up any lumps. Increase the heat to bring the mixture to a simmer, then reduce the heat and let gently bubble until the sauce has thickened slightly, about 5 minutes, stirring often. Adjust the heat as needed so that the liquid retains a rapid simmer.
Remove the pan from the heat. Stir in the vegetables, Greek yogurt, half of the cheddar cheese, mozzarella, Dijon, and thyme until smooth.
Stir in the shredded chicken and cooked rice.
Transfer the mixture to the prepared casserole dish. Sprinkle the remaining cheese over the top.
Bake the chicken and rice casserole until the cheese is melted and the casserole is hot, about 15 minutes. Turn the oven to broil, then continue cooking until the cheese is lightly browned, about 3 additional minutes, watching carefully so that the cheese does not burn. Remove from the oven and sprinkle with parsley or thyme. Let rest 5 minutes, then serve hot.