01 - Bring a large pot of salted water to a rolling boil. Add the pasta and cook 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.
03 - Heat olive oil in a large skillet over medium-high heat until shimmering. Add the seasoned chicken strips in a single layer and cook for 5 to 7 minutes, turning once, until deeply golden and cooked through to an internal temperature of 165°F. Transfer the chicken to a plate and tent loosely with foil to keep warm.
04 - Reduce the heat to medium in the same skillet. Add the butter and minced garlic, sautéing for about 1 minute until fragrant and lightly golden, being careful not to scorch the garlic.
05 - Pour in the heavy cream and whole milk, stirring to combine. Bring the mixture to a gentle simmer, then gradually whisk in the freshly grated Parmesan cheese until fully melted and the sauce is silky smooth.
06 - Add the drained pasta and cooked chicken to the skillet, tossing everything together to coat evenly in the sauce. Add the reserved pasta water a splash at a time until the sauce reaches your preferred consistency. Season with salt, black pepper, and crushed red pepper flakes if desired.
07 - Sprinkle with chopped fresh parsley, toss gently once more, and serve immediately. Garnish each plate with additional Parmesan and parsley.