Place the chicken breasts flat on a cutting board and carefully slice them in half sideways to turn 2 thick breasts into 4 thin cutlets. Mince the garlic, chop the sun-dried tomatoes into bite-sized pieces, and measure out the cream, broth, and spices before cooking. If your Parmesan is not pre-grated, grate it with any basic grater you have, or chop it very finely with a knife.
Season the chicken. Sprinkle both sides of the 4 chicken cutlets evenly with salt and black pepper.
Heat the pan. Place a large frying pan over medium-high heat. Add the olive oil and butter, then let the butter melt and gently bubble.
Cook the chicken. Add the chicken cutlets to the hot pan and cook without moving them for 4–5 minutes, until the bottom is golden brown. Flip and cook the other side for another 4–5 minutes, until cooked through and no longer pink inside.
Remove the chicken. Transfer the cooked chicken to a plate and let it rest. Do not clean the pan, because the brown bits and juices will flavor the sauce.
Cook the garlic. Reduce the heat to medium-low. Add the minced garlic to the pan and stir for about 1 minute, until fragrant. Do not let it turn dark or bitter.
Make the sauce base. Pour in the chicken broth and scrape the bottom of the pan with a spatula or spoon to loosen the browned bits.
Add the cream and flavorings. Stir in the heavy cream, Parmesan, chopped sun-dried tomatoes, dried oregano, and chili flakes if using.
Thicken the sauce. Let the sauce gently bubble for 2–3 minutes, stirring occasionally, until it thickens slightly.
Bring it all together. Return the chicken and any juices from the plate to the pan. Spoon the sauce over the chicken and let everything warm together for 1 minute.
Finish and serve. Turn off the heat. Add fresh basil or spinach if using, then serve immediately.