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 both sides. Heat the olive oil in a large skillet over medium-high heat until it shimmers. Arrange 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 to a plate and tent loosely with foil.
03 - Reduce the skillet heat to medium. Add the butter and allow it to 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 milk, stirring to combine. Bring the mixture to a gentle simmer — do not let it boil. Gradually whisk in the 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 drained pasta and seared chicken to the skillet, tossing with tongs to coat everything evenly in the sauce. If the sauce seems 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 using.
06 - Sprinkle with the chopped parsley, toss gently one final time, and divide among warmed plates. Garnish each portion with additional Parmesan and a few parsley leaves. Serve immediately while the sauce is at its creamiest.