Preheat the oven to 375°F (190°C). Lightly spray a 9x13-inch baking pan with cooking spray.
Pour 1/2 cup of the salsa into the bottom of the prepared pan and spread it evenly.
Divide the 9 tortillas into three groups of 3. Divide the cooked chicken (or cooked lean ground turkey), the refried beans, the cooked brown rice, and the cheddar cheese into three equal portions.
Make the first layer: arrange 3 tortillas in the pan so they slightly overlap. Evenly spread one-third of the cooked chicken (or turkey) over the tortillas, then one-third of the refried beans, then one-third of the cooked brown rice (about 2/3 cup), and top with one-third of the cheddar (about 1/2 cup).
Make the second layer: place the next 3 tortillas, then repeat with the next one-third portions of chicken (or turkey), refried beans, cooked rice, and cheddar.
Make the top layer: place the final 3 tortillas. Spread the remaining 1 cup of salsa over the top tortillas and sprinkle with the remaining cheddar (the final one-third).
Cover the pan tightly with foil and bake for about 30 minutes, or until the casserole is heated through and the cheese is bubbly.
Remove the foil and let the casserole rest 5 minutes. Serve with dollops of the Greek yogurt on top or on the side, and sprinkle the chopped cilantro over the casserole just before serving.