Melt the butter and olive oil together in a large ovenproof skillet over medium heat. Add the sliced onions along with sugar and salt, cooking gently and stirring now and then until the onions turn a rich golden brown, about 20-25 minutes. In the final couple of minutes, fold in the minced garlic and cook until fragrant.
Add the orzo to the caramelized onion mixture and stir it for about 2 minutes, allowing it to toast lightly. Mix in the shredded chicken, dried thyme, black pepper, and Worcestershire sauce or balsamic glaze if using.
Pour in the chicken broth and heavy cream, then bring the mixture to a gentle simmer. Lower the heat, cover the skillet, and let it simmer for 8 to 10 minutes, stirring occasionally until the orzo is cooked through and tender.
Remove the cover and stir in 1 cup of the shredded mozzarella and the Parmesan cheese until they melt completely into a creamy sauce.
Preheat your oven to 375°F (190°C). Sprinkle the remaining ½ cup mozzarella evenly over the top of the casserole. Bake uncovered for 10 to 15 minutes until the cheese on top is bubbly and has developed a golden color.
Allow the casserole to rest for 5 to 10 minutes before serving; garnish with fresh thyme or parsley if desired.