Heat oil in a skillet with a lid OR in a large pot over high heat. Add the garlic and onion, and cook for 3 minutes stirring frequently until the onion is translucent. Add rice and stir until translucent.
Add the chicken broth, milk, salt, and pepper to the skillet. Place the lid on the pan and turn the heat down to medium-low. Cook for 8-12 minutes (or until the rice is almost done).
Add shredded over the rice chicken then add the broccoli on top. Don't stir! Place the lid back on the pan and cook until the liquid is evaporated and the broccoli is just cooked, about 5 more minutes.
Remove from stove and take off the lid. Sprinkle all three kinds of cheese over the dish, place the lid back on again, and let everything sit for 5 minutes. The residual heat will melt the cheese while the rice rests. Serve warm.
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Notes
If you would like to substitute brown rice for white rice, you will need to precook the brown rice according to package directions. Brown rice has more than double the cooking time of white rice and will not work in this recipe as written.