Prepare the chicken breasts: Starting from the thickest end, slice each chicken breast in half horizontally to end up with a total of 8 cutlets. Season both sides with salt and pepper, to taste.

Spread the flour in a shallow dish. Dredge each cutlet in flour, shaking gently to remove excess and transfer to a plate.

Heat a large skillet over medium-high heat. Add 2 tablespoons butter and 1 tablespoon oil, and let the butter melt. Add 4 cutlets to the pan and cook until golden brown, 4 to 5 minutes per side. Transfer to a plate and repeat with remaining cutlets.

In the same skillet, melt remaining 2 tablespoons butter. Add onion and cook until translucent, about 2 minutes. Add garlic, red pepper flakes, and tomato paste; stir and cook until fragrant, about 1 minute. Pour in the wine to deglaze, scraping up brown bits, and let reduce by half, about 2 minutes.

Stir in chicken stock, sun-dried tomatoes, and Italian seasoning; bring to a simmer. Add chicken and any accumulated juices back into the pan and cook for 5 minutes. Stir in cream and parmesan; let simmer for another 5 minutes.

Top with parsley and extra parmesan, and serve alongside mashed potatoes or short cut pasta.