Heat a large pot of water on the stove to cook the fettuccine while preparing the Alfredo sauce.
Pour 1 cup of milk, 1/2 cup of the cream and the butter in a medium saucepan. Cook over medium heat until the mixture simmers. Reduce the heat to medium-low and simmer gently until the mixture is reduced to about 3/4 cup, about 20 minutes. Off heat, stir in the remaining 1/2 cup of cream, salt and pepper.
Meanwhile, add the pasta to the boiling water and cook until al dente, according to package directions. Reserve 1/4 cup of the pasta cooking water then drain the pasta. Return the fettuccine back to the large pot and add 2 tablespoons of the reserved pasta water. Toss to coat and cover to keep warm.
Return the cream mixture to simmer over medium-high heat, then reduce to low and add the Parmesan cheese. Cook over low heat until the Parmesan is melted, about 2 minutes. Add the remaining 2 tablespoons of pasta water to the Alfredo sauce and pour over the cooked fettuccine. Toss the pasta and Alfredo sauce using tongs to mix. The sauce may seem a little thin but will thicken as it cools.
Divide the fettuccine between 4 warm bowls and top with 1/2 a sliced chicken breast on each serving. Serve immediately garnished with chopped parsley leaves, fresh ground pepper and more Parmesan if desired.