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 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 until shimmering. Add the chicken in a single layer and cook for 5 to 7 minutes, turning occasionally, 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 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 burn the garlic. Pour in the heavy cream and whole milk, stirring to combine. Bring the mixture to a gentle simmer, then gradually whisk in the Parmesan cheese until fully melted and the sauce is silky smooth.
04 - Add the drained pasta and seared chicken to the skillet, tossing well to coat everything evenly in the sauce. Slowly incorporate 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. Sprinkle with chopped parsley, toss gently once more, and serve immediately with additional Parmesan and parsley for garnish.