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If using chicken breasts, start from the thickest end and slice each chicken breast in half horizontally so you end up with a total of 6 cutlets (see Tip). Season both sides of the chicken cutlets well with salt and pepper.
Scatter the flour on a large plate and coat the cutlets, shaking off the excess. Transfer the cutlets to a sheet pan or large plate in a single layer.
Heat the oil in a large pan over medium-high. Once hot, reduce the heat to medium and add the butter. As soon as it melts, add the cutlets and cook until golden on one side, about 5 minutes. Flip the chicken and cook the other side until golden, 4 to 5 minutes. Do this in batches, if needed, adding more oil, if needed. Transfer the cutlets to a plate or sheet pan.
Reduce the heat to low, add the garlic and cook, stirring often, until fragrant, 1 to 2 minutes. Add the tomato paste, stirring until the color deepens, about 2 minutes. Add the oregano and red-pepper flakes, to taste.
Increase the heat to medium, add the stock and bring to a simmer, scraping up any bits from the bottom of the pan, until the liquid is reduced by half, about 5 minutes.
Add 1/2 cup of the cream and warm through, stirring, until it thickens slightly, about 3 minutes. Watch the cream closely, reducing the heat if necessary, to maintain a gentle simmer. Stir in the Parmesan and the tomatoes. Add more cream, if you like, and season the sauce.
Place the chicken back in the pan to warm through, about 4 minutes. Serve with crusty bread to sop up all the juices as well as tangy green salad to balance out the sauce's richness. Or, try it over pasta, rice or polenta.