Cut each chicken breast horizontally so it's half as thick as before, resulting in 4 thinner pieces. Pat dry with a paper towel. Generously season both sides of each chicken piece with salt and pepper.
Heat a wide pan over medium to medium-high heat until hot. Add olive oil to cover the pan.
Add chicken breasts to the pan in a single layer. Cook until each side is golden brown, 4 to 5 minutes per side. When the chicken is cooked through, transfer to a plate. Work in batches if your pan is not large enough to hold them all at once.
Reduce to low heat. Add heavy cream, butter, and parmesan cheese to the pan. Stir until the butter and cheese has melted, scraping up any bits that have stuck to the pan.
Increase to medium heat and bring to a simmer. Add spinach and stir until it has wilted.
Add minced garlic and oregano, stirring until well-mixed.
Transfer the chicken back to the pan, leaving any liquid on the plate, and simmer for a minute. Top with sun-dried tomatoes. Turn off the heat.
Serve while hot, and optionally top with additional grated parmesan cheese.