01Preheat the oven to 375 F. Coat a 9 x 13-inch casserole dish with cooking spray.
02In a large pot, melt the butter over medium heat. Add the onions and sauté until translucent and tender, about 5 minutes. Add the garlic and thyme and continue to sauté for an additional minute or until the garlic is fragrant.
03Add the broth and rice. Stir to combine. Bring to a boil, then reduce to a simmer. Cover with a lid and simmer over low heat for 15 minutes.
04Turn off the heat, leaving the lid on the pot, and let sit for 10 minutes; do not stir.
05Stir into the rice the cream of chicken soup, milk, chicken, ham, cheddar cheese, Parmesan cheese, sour cream, and Dijon mustard.
06Transfer the mixture to the prepared casserole dish. Evenly top the casserole with slices of Swiss cheese. Sprinkle with breadcrumbs. Bake for 10 to 15 minutes until the cheese on the top has melted and the breadcrumbs are golden brown.