Preheat the oven to 400 degrees.
Melt the butter and olive oil over medium high heat. Add the chopped onion, and cook until translucent about 5 minutes. Add the minced garlic. Cook for an additional minute until the garlic becomes fragrant. Add the diced chicken and cook until the chicken is no longer pink. (it will continue to cook later, so don't worry about cooking it through)
Add the flour to the onion/chicken mixture. Stir well and cook for 1 minute. Slowly whisk in the milk and cream until fully incorporated. Cook until the mixture begins to thicken.
Add the uncooked rice, white wine, thyme and garlic powder. Bring to a gentle simmer and turn the heat down to low. Cook for 10-15 minutes, until the rice begins to soften. Next add the chopped broccoli and mix well. Add 1 cup of the gouda, 1 cup of cheddar and mix well. Season with salt and pepper to taste. Add the remaining 1 cup of cheddar on top.
Add to the oven and cook until the cheese begins to melt and becomes bubbly, about 8-10 minutes. Enjoy!!