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. Arrange the chicken in an even layer and cook for 5 to 7 minutes, turning once or twice, until deeply golden and cooked through to an internal temperature of 165°F. Transfer to a plate and tent loosely with foil.
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 fragrant but not browned. Pour in the heavy cream and whole milk, stirring to combine. Bring the mixture to a gentle simmer.
04 - Gradually whisk in the freshly grated Parmesan cheese, adding a handful at a time and allowing each addition to melt smoothly before adding more. Continue whisking until the sauce is velvety and free of lumps.
05 - Add the drained pasta and seared chicken to the skillet, tossing everything together with tongs until evenly coated. Splash in the reserved pasta water, a tablespoon 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, and serve right away with additional Parmesan and parsley for garnish.