In a large skillet, heat olive oil and butter over medium heat. Add the thinly sliced onions along with a pinch of salt and sugar. Cook slowly for about 20-25 minutes until golden brown.
Push the caramelized onions to the side and add the chunks of chicken. Season with salt, black pepper, dried thyme, and oregano. Sear the chicken for about 2-3 minutes on each side until browned.
Stir in minced garlic and let it cook for an additional minute.
Sprinkle the uncooked rice over the chicken and onion mixture. Pour in the beef broth and stir lightly to combine. Bring to a boil.
Reduce heat to low, cover tightly, and let it simmer for about 18-22 minutes or until the rice is cooked through and has absorbed most of the liquid.
If desired, layer in shredded cheese after turning off the heat, cover again, and let it melt for about five minutes.
Garnish with fresh chopped parsley before serving.