Season chicken with salt and pepper. Heat oil in a skillet over medium-high heat and brown breasts for 2 minutes per side, then transfer to a 5-quart slow cooker.
Add garlic, sun-dried tomatoes, broth, oregano and red pepper flakes. Cover and cook on low for 2 1/2 to 3 1/2 hours, checking from 2 1/2 hours and removing breasts as they reach 165°F.
Cover cooked chicken on a clean plate. Turn the cooker to high and stir in cream. Mix cornstarch with cold water and stir the slurry into the sauce.
Cover for 15 to 20 minutes until the sauce bubbles lightly and thickens. Switch off the heat and stir in Parmesan until smooth.
Return the whole breasts to the sauce briefly to coat them. Serve one per person with sun-dried tomato strips and cream sauce spooned over the chicken.