01 - Bring a large pot of generously salted water to a rolling boil. Add the pasta and cook according to package directions until al dente. Reserve 1/2 cup of 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 both sides. Heat the olive oil in a large skillet over medium-high heat until shimmering. 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 to a plate and tent loosely with foil.
03 - Reduce the heat under the same skillet to medium. Add the butter and allow it to melt, then stir in 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 — avoid a full boil. Gradually whisk in the freshly grated Parmesan cheese a handful at a time, allowing each addition to melt smoothly before adding more. Continue whisking until the sauce is silky and homogeneous.
05 - Add the drained pasta and seared chicken back into the skillet. Toss everything together so the sauce coats each strand or piece evenly. Splash in the reserved pasta water a little 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 - Sprinkle with chopped fresh parsley and toss gently one final time. Transfer to warm serving plates or bowls, garnishing with additional shaved Parmesan and parsley. Serve immediately while the sauce is still creamy and hot.