Set a 6-quart Instant Pot to Saute and add the butter; let it melt.
Meanwhile, place the smoked paprika, garlic powder, onion powder, mustard powder, salt, and pepper in a large bowl or zip-top bag. Add the diced chicken, seal, and shake to coat the chicken evenly with the seasonings.
Add the seasoned chicken to the melted butter and sauté uncovered for about 2 minutes, stirring and flipping frequently until the outside is lightly browned.
Add the uncooked penne to the pot, pour the chicken broth over the pasta, and stir to combine so the pasta is submerged evenly.
Secure the lid and set the valve to sealing. Cook on High Pressure for 3 minutes (the pot will take several minutes to come to pressure).
When the cook time ends, perform a quick release of the pressure, then carefully open the lid using oven mitts and a utensil if needed to avoid steam.
Turn the Instant Pot back to Saute, add the cream cheese and press it into the hot pasta so it begins to melt; replace the lid and let sit for 2 minutes.
Turn the pot off, open the lid, add the broccoli, stir to combine, then replace the lid and let the pot sit, off, for 5 minutes to steam the broccoli.
Open the lid, add the heavy cream, Colby Jack, and Parmesan; stir continuously until the cheeses melt and the sauce is smooth. Taste and adjust seasoning if desired.
Serve topped with additional Parmesan, red pepper flakes, and fresh basil if using.