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 black pepper. Heat 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. Transfer to a plate and tent loosely with foil.
03 - Reduce the skillet heat to medium. Add the butter and let it melt, then stir in the minced garlic. Sauté for about 1 minute, stirring constantly, until the garlic is fragrant and lightly golden.
04 - 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. Continue whisking until the cheese has fully melted and the sauce is smooth and velvety.
05 - Add the cooked pasta and seared chicken to the skillet, tossing everything together so the sauce coats each strand or piece evenly. Splash in the reserved pasta water a little at a time until the sauce reaches your preferred consistency. Season 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.