Place a rack in the center of the oven and preheat to 350°F. Lightly grease or spray a casserole dish.
In a large skillet, over medium heat, melt one tablespoon of butter with a tablespoon of olive oil.
Once the butter melts, add one small yellow onion, one teaspoon of salt, and one-half teaspoon of pepper. Cook for five minutes, or until the onion is translucent. Stir in two teaspoons of minced garlic and let cook for one minute.
Sprinkle three tablespoons of Bisquick over the top. Stir constantly until the flour turns brown.
Pour in one can of chicken broth and one cup of milk, stirring constantly to prevent any lumps.
Bring the mixture to a simmer, then reduce the heat until the sauce has thickened, stirring often.
Remove the pan from the heat, and stir one bag of mixed frozen vegetables, one cup of plain greek yogurt, half of the cheddar cheese, one-half cup of mozzarella, one teaspoon of dijon mustard, and one-half teaspoon of thyme until smooth.
Stir in three cups of shredded chicken and three cups of cooked rice.
Transfer the mixture to the prepared casserole dish. Sprinkle the remaining cheese over the top.
Bake the casserole for fifteen minutes, or until the cheese has melted.
Remove from the oven and sprinkle with parsley. Let rest for five minutes and serve.