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. Before draining, reserve 1/2 cup of the starchy pasta water. Drain the pasta and set 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 5–7 minutes, turning occasionally, until golden brown and cooked through to an 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 minced garlic, sautéing for about 1 minute until fragrant and lightly golden. Pour in the heavy cream and milk, stirring to combine. Bring the mixture to a gentle simmer — do not let it boil.
04 - Gradually whisk in the freshly 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 lump-free.
05 - Add the drained pasta and seared chicken back into the skillet, tossing everything together so the sauce coats each strand or piece evenly. 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.
06 - Sprinkle with chopped fresh parsley and toss gently. Serve immediately in warmed bowls, garnished with additional Parmesan and parsley.