Heat 2 tablespoons of oil over medium-high heat in a large non-reactive saute pan, such as enamel coated cast-iron or stainless-steel pan. Season the chicken generously with salt and pepper, to taste. Make sure to season both sides of the chicken.

When the pan is hot and the oil is shimmering, add the chicken to the skillet in a single layer. Cook for 3-4 minutes on each side undisturbed or until just about cooked through. Cook time will depend on the thickness of the breast. For smaller pieces of chicken, the time may be more like 2-3 minutes per side. Remove to a plate and cover lightly with foil to keep warm.

Reduce the heat to medium. Add the remaining 1 tablespoon of oil. When the oil is hot, add tomatoes, garlic, and oregano with a pinch of salt. Cook tomatoes, stirring occasionally until softened, about 2-3 minutes.

Stir in chicken broth, heavy whipping cream, red pepper flakes, and sugar.  Season with salt and pepper to taste. Bring to a simmer, while scraping down the bottom.

Return chicken to the pan and simmer with sauce, partially covered, for about 1-2 minutes or until sauce slightly thickens. Stir in Parmesan cheese and basil. Serve with pasta, rice, or crusty bread to soak up the delicious sauce.