Pat the chicken dry and season with salt and black pepper.
Heat the olive oil in a large skillet or saute pan over medium-high heat.
When the oil is hot and shimmering, place the chicken breasts in the hot oil and cook until nicely browned on the first side, approximately 4-5 minutes. Flip the chicken and cook another 4-5 minutes or until the chicken is cooked through to 165 degrees in the thickest part. Transfer to a plate and tent with aluminum foil to keep warm.
Add the shallot, reduce heat to medium-low and cook for 2-3 minutes or until softened. (If necessary, add a little water or chicken broth to keep the shallot from burning. The shallot will take on the color of the fond, but you don't want it to brown or burn.)
Add the garlic and cook briefly until fragrant, approximately 10 seconds.
Add the white wine and bring to a boil, stirring and scraping up all the browned bits. Cook for 1-2 minutes or until the wine is slightly reduced.
Add the chicken broth, heavy cream, sun-dried tomatoes, oregano and crushed red pepper flakes.
Bring to a boil, reduce heat to medium and simmer for 5 minutes or until slightly reduced and thickened.
Stir in the Parmesan cheese until it's melted and blended into the sauce. (The sauce should now be thickened enough to the point where you can run a spatula or spoon through it and the path will not immediately fill in.)
Stir in the fresh basil.
Season with salt and black pepper to taste. Adjust the flavor balance with 1 to 2 tablespoons of additional dry white wine if the sauce tastes too sweet to you.
Slice the chicken and carefully return the chicken to the skillet and simmer briefly on low to bring it back up to the safe temperature of 165 degrees.
Serve over angel hair pasta if desired. Sprinkle with additional Parmesan cheese if desired.