If necessary, cover chicken breasts with plastic wrap and use a mallet to lightly pound them until ½-inch thick. Season with salt and freshly ground black pepper on both sides.
Preheat oven to 375 degrees F.
Cook pasta according to package directions.
Meanwhile, add olive oil in a large oven-safe skillet and place over MEDIUM-HIGH. When oil is good and hot add the chicken and cook for 2 to 3 minutes per side, until nicely browned. Chicken does not need to be cooked through at this point. Transfer the browned chicken to a plate or large piece of foil to rest. Do this in batches, if necessary, to avoid over-crowding the skillet.
Reduce the heat to MEDIUM, add the shallots and garlic to the skillet. Cook, stirring, for a minute or two. Add the white wine and use a spoon to scrape up any browned bits from the bottom of the skillet as the wine comes to a simmer. Allow wine to simmer at a low boil for 3 to 4 minutes until slightly reduced. Add the chicken broth, lemon juice, and red pepper flakes. Increase heat to MEDIUM-HIGH and bring sauce to a low boil. Allow to simmer for about 10 to 15 minutes until thickened and reduced.
Remove the skillet from the heat and add the butter and stir until completely melted. Add the heavy cream and stir to combine. Transfer the browned chicken to the skillet and drizzle the sauce over the top. Transfer the skillet in the oven for 8 to 10 minutes or until the chicken is completely cooked through. Remove from oven and top with 2 tablespoons chopped basil.
Drain pasta, reserving ½ cup pasta water. Place pasta in a large bowl and add cheese and remaining basil and toss to combine, adding enough reserved pasta water to moisten pasta. Serve with chicken and and spoonful of sauce.