Bring a large pot of salted water to a boil over high heat. Cook the pasta until it is al dente (about 2-3 minutes less than the cooking time listed on package directions).
As the pasta cooks, melt the butter in a large 12-inch skillet or Dutch oven set over medium heat. Sprinkle the melted butter with the flour and use a whisk to combine (it will form a thick paste). Cook the paste (roux) for about 30 seconds, stirring often.
Add the minced garlic, chicken broth, and lemon juice. Whisk continuously as it comes to a gentle simmer. Allow it to simmer/bubble for about 30 seconds.
Still whisking, slowly stir in the cream and lemon zest until fully combined with the flour mixture. Bring to a gentle simmer then reduce the heat to medium-low. Cook over medium-low heat for about 2 minutes, or until it has thickened slightly.
Stir in the grated parmesan cheese and shredded chicken. Season with salt and pepper, to taste. Remove from heat.
Reserve at least 1-2 cups of hot pasta water before draining the pasta. Add the drained pasta to the sauce and toss to coat. Add some of the reserved pasta water to the pan, if the sauce becomes too thick (the starch from the pasta thickens the sauce).
Season with additional salt, pepper, and lemon zest, to taste, and serve garnished with additional grated parmesan cheese and fresh parsley, if desired.