Place chicken breasts on cutting board. Generously season tops of chicken with salt and pepper to taste, then flip chicken breasts over and season other side. Set cutting board aside.
Add chicken broth to medium mixing bowl. Slowly add cornstarch, whisking constantly to fully incorporate cornstarch into broth. Continue whisking until slurry mixture is completely smooth, with no lumps of cornstarch remaining. Once smooth, pour slurry into slow cooker.
Add heavy cream, garlic, oregano, dried oregano, salt, pepper, crushed red pepper flakes, sun-dried tomatoes, and sun-dried tomato oil (if using) to slow cooker. Whisk all ingredients together until fully incorporated.
Place seasoned chicken breasts in slow cooker in one even layer if possible. Ladle sauce mixture over chicken breasts to cover.
Secure lid on slow cooker. Cook on HIGH 2 ½ to 3 hours, or cook on LOW 6 to 7 hours, until chicken reaches 160° Fahrenheit internally.
Once chicken reaches target temperature, carefully remove chicken breasts from slow cooker and transfer to large plate. Set chicken aside to rest 5 minutes.
Carefully remove slow cooker insert and set on trivet, stovetop, or other heat-resistant surface. Whisk sauce vigorously until smooth, then allow sauce to cool slightly.
Add room temperature parmesan to sauce and whisk until cheese has melted and is fully incorporated into smooth sauce.
Transfer chicken to serving dishes with preferred sides. Ladle parmesan sauce over tops of chicken breasts and garnish with chiffonaded basil. Serve immediately.