Add the oil, butter, and onion to a pot over medium-high heat. Sauté the onion for 3-4 minutes.
Add the garlic, Italian seasoning, and orzo. Cook for 2-3 minutes, stirring often.
Stir in the chicken broth, cream, pesto, sun-dried tomatoes, and asparagus. 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.
Stir in the cooked chicken (let it warm through for a minute or so).
Take the pot off the heat and stir in the parmesan cheese. Give it a taste, and if the pasta is firmer than you'd like or you want the sauce to be thicker/more absorbed, cover the pot and let it sit for a few minutes. If the sauce has reduced too much, add in a splash of cream. Season with salt & pepper as needed and enjoy immediately.