Melt butter and olive oil in a large oven-safe skillet over medium heat.
Add sliced onions, sugar, and salt, then cook stirring occasionally for 20 to 25 minutes until deeply golden and caramelized.
Stir in minced garlic during the last 2 minutes of cooking.
Add orzo pasta to the skillet and cook for 2 minutes to lightly toast it.
Mix in shredded chicken, dried thyme, black pepper, and Worcestershire sauce or balsamic glaze if using.
Pour in chicken broth and heavy cream; bring to a gentle simmer.
Reduce heat to low, cover, and cook for 8 to 10 minutes, stirring occasionally, until the orzo is tender.
Stir in 1 cup of mozzarella and all Parmesan cheese until melted and sauce is creamy.
Preheat oven to 375°F and sprinkle remaining 1/2 cup mozzarella on top.
Bake uncovered for 10 to 15 minutes until bubbly and golden brown.
Allow the dish to rest for 5 to 10 minutes before serving.
Garnish with fresh thyme or parsley if desired.