Add in 2 tablespoons of butter and 2 tablespoons of olive oil. Add in the garlic, onion, and roasted peppers. Cook, stirring for 2-3 minutes, then add in the spinach and 3 of the diced tomatoes. Season with salt and pepper. Cook until the spinach has wilted slightly. Stir in the cream and allow it to thicken/reduce slightly, about 3 minutes. Reduce the heat to medium and stir in the Parmesan and cream cheese. Taste and adjust seasoning (be sure to adequately season.) Remove the pan from the heat and set aside. With a slotted spoon, remove about 1 cup of the mixture to a bowl and allow it to cool slightly.
With a sharp knife, make a pocket in each chicken breast. Sprinkle the inside with a little salt and pepper. Stuff each pocket with 1/4 of the cooled mix. Seal the chicken breasts with 3 toothpicks to keep the mixture from falling out while cooking. Season the outside of the chicken with salt and pepper.
Preheat the oven to 400 degrees.
Heat another cast iron (or ovenproof) skillet over medium-high heat. Add in the remaining butter and olive oil and place the breasts smooth side down in the skillet. cook for 4 minutes, then turn. Place the skillet in the oven for 8 to 10 minutes, or until the chicken is cooked through and the filling is hot.
Reheat the creamy spinach/red pepper mixture. Using a slotted spoon, transfer some of the creamed spinach to a plate. Remove the toothpicks from the chicken and place it on top of the spinach. Spoon some of the chicken pan sauces over the chicken. Top with diced tomatoes, basil, and Parmesan shavings. Repeat with the rest.