Set a 6- or 8-quart Instant Pot to Sauté and wait until the display reads “Hot.”
Add the olive oil, then the cubed chicken; sauté 4–5 minutes until the exterior is golden and mostly cooked through, stirring frequently. Transfer the chicken to a plate.
Add the chopped onion and minced garlic to the pot and cook about 2 minutes until softened and lightly browned, stirring to prevent sticking.
Pour in 1 cup of the chicken broth and scrape the bottom of the pot to deglaze. Press Cancel to stop Sauté mode.
Spread the dry rigatoni in an even layer over the liquid, then add the remaining 2 cups of broth. Pour the tomato sauce on top without stirring.
Close the lid, set the valve to Sealing, and Pressure Cook (Manual) on High for 5 minutes.
When the cook time ends, perform a quick release by carefully turning the valve to Venting.
Remove the lid, add 1/2 cup mozzarella, 1/4 cup Parmesan, and the cooked chicken to the pot; gently stir to combine. Sprinkle the remaining 1/2 cup mozzarella and 1/4 cup Parmesan over the top.
Close the lid off the heat and let sit 5 minutes to allow the cheeses to melt, then stir gently and serve immediately.