Add the butter and olive oil to a large oven-safe skillet or Dutch oven and place over medium heat.
Add the sliced onions, sugar, and salt. Cook for 20–25 minutes, stirring occasionally, until the onions become soft, deeply golden, and caramelized. Reduce the heat if they begin browning too quickly.
Stir in the minced garlic during the final 1–2 minutes of cooking.
Add the orzo and cook for about 2 minutes, stirring frequently, to lightly toast the pasta.
Add the shredded chicken, dried thyme, black pepper, and optional balsamic glaze. Stir until everything is evenly combined.
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 to prevent the orzo from sticking. Cook until the pasta is almost tender and the sauce has started to thicken.
Preheat the oven to 375°F (190°C).
Stir 1 cup of mozzarella and all of the Parmesan into the French Onion Chicken Orzo Casserole until the cheese melts into the creamy sauce.
Sprinkle the remaining 1/2 cup mozzarella evenly over the casserole.
Transfer the uncovered skillet to the oven and bake for 10–15 minutes, or until the casserole is bubbling and the cheese is melted and lightly golden.
Let it rest for 5–10 minutes before serving. Garnish with fresh thyme or parsley if desired.