Put a large pot of water on to boil, for the pasta (I like to use fettucine).
Slice the chicken breasts in half lengthwise to create cutlets. Blot them with a paper towel to remove excess moisture, then season with salt and pepper on both sides.
On medium-low, heat the two oil and butter in a large saucepan.
Place the flour and the chicken breasts in a zip-closure plastic bag, seal it and shake the contents to coat the chicken evenly. Shake off excess.
Add the chicken to the saucepan and brown it on both sides, taking care not to crowd the chicken. Brown them on both sides and cook through; the time this takes will depend on the thickness of your cutlets, but figure about 4-6 minutes each side.
Remove the cutlets from the pan and set them aside, covered with foil or a spare pot lid.
By now your pot of water is probably boiling. Add the pasta of your choice to the water and cook according to the package instructions while you continue with the chicken dish.
Add the garlic, red pepper flakes, oregano, thyme, and half of the basil to the pan and let them bloom for about 30 seconds.
Now add the chicken broth or stock to the pan. Using a flat-edged wooden spoon (I have both the OXO beechwood and the Norpro bamboo and love them both), deglaze the pan with the broth to scrap up any brown bits (called fond) from the pan surface - it's great flavor! Bring it to a simmer.
Pour the heavy cream into the pan and add the parmesan cheese. Stir frequently until the cheese is completely melted and blended into the sauce.
Add the chicken back to the pan and allow them to simmer in the sauce until heated through, stirring gently and frequently to prevent the sauce from sticking to the pan.
Drain the pasta and arrange it on a serving platter. Top with the chicken and sauce, then garnish with the other half of the chopped basil. Serve.