Jarred alfredo sauce always tastes like it is missing something, and what it's missing is real cream and real parmesan simmered together instead of stabilizers.
This version takes the same 30 minutes as boiling pasta and opening a jar, but the sauce actually tastes like something.
Keep the cream at a gentle simmer, never a hard boil, once the parmesan goes in — high heat is what causes a grainy, broken sauce.
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Cook fettuccine in salted boiling water according to package directions until al dente. Reserve 1/2 cup pasta water, then drain.
Season chicken strips with salt and pepper. Heat 1 tbsp butter in a large skillet over medium-high heat and cook chicken for 6-7 minutes until golden and cooked through. Remove and set aside.
In the same skillet, melt remaining butter and sauté garlic for 30 seconds. Pour in heavy cream and bring to a gentle simmer. Whisk in parmesan a handful at a time until melted and smooth.
Add the drained pasta and chicken to the sauce, tossing to coat, adding reserved pasta water a splash at a time if the sauce is too thick. Top with parsley and extra parmesan.
The heat was likely too high when adding the parmesan, which can cause the proteins to seize — keep the cream at a gentle simmer, not a boil, and add cheese gradually off direct high heat.
Milk will make a thinner, less rich sauce and is more likely to separate — heavy cream's fat content is what keeps the sauce stable and silky.
The starchy water helps loosen and re-emulsify the sauce if it thickens too much once combined with the pasta, without watering down the flavor the way plain water would.