Add the oil, butter, and onion to a pot over medium-high heat. Sauté the onion for 3-4 minutes.
Add the garlic and orzo. Cook for 2-3 minutes, stirring often (this infuses the orzo with more flavor and toasts it a bit).
Stir in the chicken broth and cream. Once it starts to bubble, continue cooking for 10 minutes, uncovered, stirring fairly often. You will likely need to turn the heat down a bit (to medium or even medium-low). You want it to gently bubble vs. furiously boil, and you don't want the liquid to reduce too much before the pasta has cooked.
Take the pot off the heat, stir in the cheeses, and cover the pot for 4-5 minutes or until it has thickened up to your liking (this recipe is meant to have a lot of sauce but will thicken a little more as it cools - orzo absorbs liquid quite fast). Season with salt & pepper as needed.
Notes
Orzo is pasta. If you want to make this with rice, you may have to adjust liquid measurements and cooking time (it would be a different recipe).