Preheat oven to 400F. Combine squash, onion, olive oil, thyme, salt, and pepper to taste on a large sheet pan, tossing until evenly combined. Arrange in a single layer on sheet pan.
Bake squash and onion at 400F 20-25 minutes until vegetables are almost tender. Leave oven on and set vegetables aside to cool.
In a large high-walled skillet, melt butter over medium heat. Add garlic and saute 1 minute until fragrant. Whisk in flour and cook several minutes until golden brown. Slowly begin whisking in milk and vegetable broth, mixing constantly until smooth.
Bring sauce to a simmer over medium heat, stirring occasionally until sauce is slightly thickened and bubbly. Fold in roasted squash/onion, wild rice, and white or brown rice until evenly combined. Add sage, salt, and pepper to taste.
Scrape rice mixture into a greased 9x13 baking dish and top with cheeses. Bake casserole at 400F 25-30 minutes until bubbly and cheese is melted. Top casserole with additional thyme and serve warm. Enjoy!