Place a large, oven-safe skillet or Dutch oven over medium heat. Melt the unsalted butter and olive oil together. Add the thinly sliced yellow onions, sprinkling them immediately with the sugar and salt. Reduce the heat to medium-low and cook slowly for 20 to 25 minutes, stirring occasionally. Once the onions are deeply golden and jammy, stir in the minced garlic and cook for 1 to 2 more minutes until fragrant.
Keeping the heat on medium-low, pour the dry orzo directly into the skillet with the caramelized onions. Stir constantly for 2 minutes to lightly toast the pasta. Fold in your shredded cooked chicken, dried thyme, black pepper, and the optional Worcestershire sauce until everything is evenly combined.
Pour the chicken broth and heavy cream into the skillet, bringing the liquid to a gentle simmer. Immediately reduce the heat to the lowest setting, cover the skillet with a tight-fitting lid, and let it cook undisturbed for 8 to 10 minutes. Stir once or twice to prevent sticking, cooking until the orzo is tender and the liquid has thickened into a glossy sauce.
Remove the skillet entirely from the heat. Sprinkle 1 cup (approx: 113g) of the shredded mozzarella and the full 1/2 cup of grated Parmesan cheese into the hot pasta. Stir gently but thoroughly until the cheeses are completely melted into the creamy sauce.
Preheat your oven to 375°F (190°C). Scatter the remaining 1/2 cup (approx: 57g) of mozzarella cheese evenly across the top of the skillet. Bake uncovered on the center rack for 10 to 15 minutes, or until the cheese is bubbling furiously and forms a beautiful, golden-brown crust. Let the casserole rest for 5 to 10 minutes before garnishing with fresh parsley and serving!