Preheat oven to 375° Fahrenheit. Adjust racks as needed so skillet can sit in center of oven. Place chicken breasts on cutting board and trim as needed, then pat chicken breasts dry on all sides with paper towels.
Generously season tops of chicken breasts with salt and black pepper. Flip chicken breasts over and generously season other side with salt and black pepper. Set chicken aside.
Heat cast-iron skillet over medium-high heat. When skillet is warm, add olive oil and swirl pan to cover entire surface with oil.
When oil is hot and shimmery, place seasoned chicken breasts in skillet. Sear chicken breasts 3 to 4 minutes, or until lightly golden in spots. Flip chicken breasts over and sear 3 to 4 minutes more. Carefully transfer seared chicken breasts to plate and set aside. Note: chicken should not be fully cooked-through at this point.
Return skillet to stovetop and reduce heat to medium-low. Let temperature of skillet reduce, then add minced garlic. Sauté garlic 1 to 2 minutes, or until garlic is just fragrant.
Add broth to skillet and stir, making sure to scrape up any browned bits of chicken or garlic that may be stuck to bottom of skillet.
Move skillet to cool stovetop burner and let skillet cool slightly, then add heavy cream. Whisk or stir gently to incorporate heavy cream into chicken broth. Once liquids are blended, sprinkle grated parmesan into mixture and gently stir until cheese has melted.
Return skillet to hot burner and continue stirring sauce mixture 1 minute or so, while skillet returns to temperature. Add sun-dried tomatoes, dried oregano, and red pepper flakes to cream sauce and stir gently to incorporate ingredients. Taste sauce and adjust salt, pepper, oregano, and/or red pepper flakes as desired.
Bring sauce to constant slow simmer over medium-low heat, stirring occasionally. Once sauce is simmering, return seared chicken breasts to skillet. Flip chicken breasts 3 to 4 times to coat all sides in sauce, then transfer entire skillet into preheated oven.
Bake dish 12 to 15 minutes, then begin checking doneness of chicken with internal meat thermometer. Continue cooking chicken until meat thermometer reads 160°-162° Fahrenheit when inserted into thickest part of chicken.
Carefully remove skillet from oven and set aside. Spoon sauce from skillet over tops of chicken, then allow chicken to rest 5 minutes. Chicken will cook residually, reaching food-safe internal temperature of 165°F.
After 5 minutes, transfer chicken to serving plates. Top chicken with remaining sauce from skillet and garnish with chiffonaded basil. Serve immediately with desired sides.
