Sprinkle chicken with salt and pepper. Heat sun-dried tomato oil in a large skillet over medium heat. Add the chicken and cook, turning once, until golden brown and just cooked through. Work in batches if necessary. Remove chicken, lightly covered to keep warm and leave drippings behind in the pan.

Add sun-dried tomatoes and shallots to the pan. Cook, stirring, for a minute or two, until shallots are just beginning to become tender. Increase heat to high and add wine. Cook, scraping up any browned bits until the liquid has mostly evaporated, about two minutes.

Reduce heat to medium and stir in cream along with any juices that have accumulated with the chicken, as well as salt and pepper to taste. Cook for about two minutes, stirring often to slightly reduce the cream. Return the chicken to the pan and turn to coat with the sauce.

Serve the chicken topped with the sauce and parsley.