Preheat oven to 375 degrees.
Melt 2 Tablespoons butter and add the diced onions. Cook until softened, about 5 minutes.
Add the garlic, garlic powder, and thyme. Cook for one more minute.
Add olive oil, chicken broth, and rice. Stir to combine.
Bring to a boil and reduce to a simmer. Cover tightly and cook for 15 minutes. Do not stir.
Turn off heat, leave the lid on and let it sit for 10 minutes, do not stir. Any rice stuck on the bottom will release during this time.
Add the cream of chicken soup, sour cream, milk, Dijon mustard, cheddar + Parmesan cheese, and cooked chicken/ham. Stir to combine.
Transfer to a lightly greased 9 x 13 casserole dish. Top with Swiss cheese. Cover and bake for 15 minutes.
Combine the melted butter and breadcrumbs. Sprinkle over the top of the casserole and bake for 10 more minutes, uncovered.
Let it stand for 5 minutes prior to serving.