Melt the butter and olive oil together in a large oven-safe skillet or Dutch oven over medium heat.
Add the thinly sliced onions plus sugar and salt; cook gently, stirring occasionally, until the onions are rich golden brown and sweetly caramelized, about 20 to 25 minutes.
Stir in the minced garlic and cook for another 1 to 2 minutes to release its aroma.
Mix the orzo into the skillet and toast it lightly by stirring for around 2 minutes.
Incorporate the shredded chicken, dried thyme, black pepper, and Worcestershire sauce or balsamic glaze if you are using it; stir well to combine all flavors.
Pour in the chicken broth and heavy cream, then bring the mixture to a soft simmer.
Lower the heat to the minimum, cover the pot, and gently cook for 8 to 10 minutes, stirring now and then until the orzo is tender and creamy.
Stir in 1 cup of mozzarella cheese along with the grated Parmesan until the cheeses melt smoothly into the sauce.
Preheat your oven to 375°F (190°C).
Sprinkle the remaining ½ cup mozzarella evenly on top and bake uncovered for 10 to 15 minutes until the surface is golden and bubbly.
Allow the casserole to rest for 5 to 10 minutes after baking to set.
Serve warm, garnished with fresh herbs like thyme or parsley if desired.