In a large, ovenproof skillet or Dutch oven, warm the butter and olive oil over medium heat. Add the sliced onions, sugar, and salt. Sauté for 20–25 minutes, stirring occasionally, until onions turn deep golden brown and caramelized. Add garlic during the final 1–2 minutes of cooking time and stir to combine.
Introduce the orzo to the pan and cook for about 2 minutes, stirring frequently to lightly toast the pasta. Mix in the shredded chicken, thyme, black pepper, and Worcestershire sauce or balsamic glaze if using.
Pour in the chicken broth and cream. Bring the mixture to a low simmer. Lower the heat, cover the pan, and cook for 8–10 minutes, stirring from time to time, until the orzo becomes tender and the mixture thickens slightly.
Fold in 1 cup of mozzarella cheese and the Parmesan cheese. Stir until both cheeses melt and the mixture becomes creamy and smooth.
Preheat the oven to 375°F (190°C). Scatter the remaining ½ cup mozzarella evenly over the top of the dish. Place the skillet or Dutch oven in the oven and bake uncovered for 10–15 minutes, or until the top is bubbling and lightly golden.
Remove from the oven and allow the dish to rest for 5–10 minutes before serving. Optionally, garnish with fresh thyme leaves or chopped parsley.