In a large oven-safe skillet or Dutch oven, melt 2 tablespoons of butter with 1 tablespoon of olive oil over medium-low heat. Add 3 sliced yellow onions, 1 tablespoon of sugar, and a pinch of salt. Cook the onions gently, stirring frequently, for about 30-40 minutes until they turn golden and deeply caramelized. Add 2 minced garlic cloves during the last minute for extra flavor.

While the onions are caramelizing, take 2 cups of cooked shredded chicken and season it with salt and black pepper to taste. In a separate skillet, heat a splash of olive oil over medium heat, then add the seasoned chicken. Brown the chicken for about 5-7 minutes until golden.

After the onions have caramelized, pour in 1 cup of low-sodium chicken broth and gently scrape up the brown bits from the bottom of the skillet using a wooden spoon. Stir in 1 cup of orzo pasta, 1 cup of heavy cream, 1 teaspoon of dried thyme, and the browned chicken. Let this simmer on low heat for about 5-7 minutes until the orzo starts to absorb some of the liquid.

Once the orzo has absorbed some liquid, mix in half of the 1 cup of shredded mozzarella cheese until melted and combined. Sprinkle the remaining cheese over the top, then transfer the skillet to a preheated oven at 375°F (190°C). Bake uncovered for 20-25 minutes, or until the cheese is golden and bubbly.

After baking, remove from the oven and let the casserole sit for about 5-10 minutes to thicken slightly. Garnish with fresh thyme or chopped parsley if desired, and serve.