01 - Bring a large pot of salted water to a rolling boil. Cook the fettuccine or penne according to package directions until al dente. Reserve 1/2 cup of the starchy pasta water before draining. Set the drained pasta aside.
02 - While the pasta cooks, season the chicken strips generously with salt and freshly ground black pepper on all sides. Heat the olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook for 5 to 7 minutes, turning once, until golden brown and cooked through to an internal temperature of 165°F. Transfer the chicken to a plate and tent loosely with foil to keep warm.
03 - Reduce the skillet heat to medium. Add the unsalted butter and allow it to melt, then add the minced garlic. Sauté for about 1 minute, stirring constantly, until the garlic is fragrant and lightly golden — do not let it brown.
04 - Pour in the heavy cream and whole milk, stirring to combine. Bring the mixture to a gentle simmer — do not allow it to reach a full boil. Gradually whisk in the freshly grated Parmesan cheese, a handful at a time, until fully melted and the sauce is smooth and velvety.
05 - Add the drained pasta and seared chicken to the skillet, tossing with tongs to coat everything evenly in the sauce. Pour in the reserved pasta water a splash at a time, tossing between additions, until the sauce reaches your preferred consistency. Season to taste with salt, black pepper, and crushed red pepper flakes if desired.
06 - Remove from heat. Sprinkle with the chopped fresh parsley and toss gently one final time. Serve immediately in warm bowls, garnished with additional shaved Parmesan and a scattering of fresh parsley leaves.