Caramelize the onions: Melt the butter and olive oil in a large oven-safe skillet or Dutch oven over medium heat. Add the onions, sugar, and salt. Cook for 20–25 minutes, stirring occasionally, until deeply golden and caramelized. Add the garlic during the final 1–2 minutes.
Toast the orzo: Stir in the orzo and cook for about 2 minutes, stirring frequently, until lightly toasted.
Add the chicken and seasonings: Stir in the shredded chicken, dried thyme, black pepper, and Worcestershire sauce or balsamic glaze, if using.
Add the liquids: Pour in the chicken broth and cream. Bring the mixture to a gentle simmer, then reduce the heat to low. Cover and cook for 8–10 minutes, stirring occasionally, until the orzo is tender.
Make it cheesy: Stir in 1 cup of the mozzarella and all of the Parmesan until the cheeses melt into the creamy orzo.
Bake: Preheat the oven to 375°F (190°C). Sprinkle the remaining ½ cup mozzarella evenly over the top. Bake uncovered for 10–15 minutes, until bubbling and golden.
Rest and serve: Let the casserole rest for 5–10 minutes before serving. Garnish with fresh thyme or parsley if desired.