Season the chicken evenly with salt and pepper. Heat the sun-dried tomato oil in a large skillet over medium-high heat. Add the chicken and sear until golden on both sides — about 4 minutes per side for breasts or 6 for thighs. Remove and set aside so they stay juicy while you work on the sauce.
Reduce the heat to medium and add minced garlic. Stir for about 15 seconds until fragrant, then pour in the white wine (or broth). Scrape up the browned bits from the pan — this is where the flavor hides. Let it simmer for a minute so the sharpness of the wine mellows.
Whisk in Dijon mustard, chicken broth, heavy cream, Parmesan, and chopped sun-dried tomatoes. Let it simmer gently for 3 minutes until it thickens slightly. Taste and adjust seasoning if needed — a pinch more salt or pepper can really bring it together.
Return the chicken to the skillet, spooning sauce over each piece. Let it cook for one more minute so the flavors fully blend. Sprinkle with fresh basil before serving for a burst of freshness.