Pat the chicken breast fillets thoroughly dry with paper towels. Season both sides with salt and black pepper, then distribute the dried Italian seasoning evenly over each piece, ensuring the surface is well coated.

Warm the unsalted butter and olive oil in a large skillet over medium-high heat. Alternatively, use the browning setting on the slow cooker if available. Add the seasoned chicken, placing skin-on fillets skin-side down first, and sear for approximately 2 to 3 minutes on each side until golden. If using a separate skillet, transfer the browned chicken to the slow cooker.

Turn the chicken as needed and pour in either the white wine and water mixture or the chicken broth. Sprinkle the sweet paprika and garlic powder over the chicken. When using wine and water, add the chicken bouillon granules as well. Scatter the drained, chopped sun-dried tomatoes around the chicken.

Cover the slow cooker and cook on LOW for 90 minutes to 2 hours. The chicken is ready when it is tender and the thickest portion reaches an internal temperature of 165°F (75°C).

For a thicker consistency, stir together the cornstarch and cold water until smooth. Add this slurry to the sauce approximately 10 minutes before the end of the cooking time, then allow the sauce to continue cooking until slightly thickened.

Once the chicken is fully cooked, move the pieces toward one side of the slow cooker. Stir the sour cream or heavy cream into the cooking liquid until completely incorporated and smooth.

Add the grated Parmesan cheese along with the chopped flat-leaf parsley or fresh basil. Stir gently until the cheese blends into the creamy sauce.

Serve the Crock Pot Marry Me Chicken hot over rice, noodles, or mashed potatoes. Spoon a generous amount of the creamy sun-dried tomato sauce over each portion and finish with additional fresh herbs if desired.