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 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 once, 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 let it 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 over medium heat. Gradually whisk in the grated Parmesan cheese a handful at a time, allowing each addition to melt smoothly before adding more. Continue whisking until the sauce is velvety and homogeneous.
05 - Add the cooked pasta and seared chicken to the skillet, tossing everything together so the sauce coats each strand or piece evenly. If the sauce seems too thick, loosen it with the reserved pasta water, adding a tablespoon at a time until it reaches the desired silky consistency. Season to taste with salt, black pepper, and crushed red pepper flakes if desired.
06 - Sprinkle with the chopped fresh parsley and toss gently one last time. Plate immediately, garnishing each serving with additional Parmesan and a few parsley leaves. Serve hot.