On a cutting board, cover each chicken breast with plastic wrap and, using a mallet or heavy pan, pound each breast to an even thickness. Season the chicken breasts on both sides with kosher salt and black pepper. Lightly dust each with flour.
In a large skillet, heat the olive oil over medium-high heat. Cook the chicken breasts for approximately 4 minutes per side, until evenly browned and cooked through. Remove to a separate plate and cover.
Into the same skillet, add the butter and reduce the heat to medium. Add the sliced onions and mushrooms and cook, stirring often, until softened, about 7 minutes. Stir in the minced garlic and cook 3 minutes longer. Season with salt and black pepper.
Bring a large pot of water to a boil.
Bring the heat up to medium-high. Pour in the white wine and simmer until reduced by half, about 5 minutes. Cook the pasta according to package instructions. Reduce the heat back to medium and stir in the cream and Parmesan cheese. When the sauce is bubbling, return the chicken to the pan.When the pasta is cooked, stir in the pasta, plus ⅓ cup pasta water, into the sauce. Simmer 1-2 minutes more until the sauce is slightly thickened. Serve immediately.