Preheat the oven to 375°F (190°C).
In a medium saucepan, combine the uncooked brown rice and chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 35 minutes or until rice is tender and liquid is absorbed.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add chopped onion and minced garlic, sauté for 3-4 minutes until softened.
Add the chicken breast pieces to the skillet. Cook for 6-8 minutes, stirring occasionally, until chicken is cooked through and no longer pink inside.
Steam the broccoli florets until just tender, about 4-5 minutes, then drain well.
In a large mixing bowl, combine the cooked rice, cooked chicken and onion mixture, steamed broccoli, Greek yogurt, low-fat milk, dried thyme, salt, and black pepper. Stir until thoroughly mixed.
Fold in 3/4 cup of the shredded cheddar cheese.
Transfer the mixture to a 9×13 inch baking dish and spread evenly.
Sprinkle the remaining 1/4 cup shredded cheddar cheese evenly over the top.
Bake uncovered in the preheated oven for 15 minutes, or until the casserole is heated through and cheese is melted and bubbly.
Remove from oven and let rest for 5 minutes before serving.