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. Add the chicken in a single layer and cook for 5 to 7 minutes, turning once, until golden brown and cooked through (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, melt the butter and add the minced garlic. Sauté for about 1 minute, stirring constantly, until the garlic is fragrant and lightly softened — 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 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 cooked pasta and seared chicken back into the skillet. Toss everything together until evenly coated in the sauce. If the sauce is too thick, add the reserved pasta water a splash at a time until it 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. Serve immediately in warmed bowls, garnished with additional Parmesan and parsley.