Bring a large pot of salted water to a boil for the pasta.
Sprinkle the chicken with some salt.
Heat a large skillet over medium-high heat until hot, then add the oil.
Working in 2 batches, brown the chicken until golden but not cooked through, 2 to 3 minutes per batch.
Remove the chicken to a plate.
Melt 4 tablespoons of the butter in the skillet.
Add the garlic and red pepper flakes and cook until the garlic just begins to turn golden at the edges, 30 seconds to 1 minute.
Add the wine, bring to a simmer and cook until reduced by half, about 2 minutes.
Remove from the heat.
Meanwhile, cook the pasta until very al dente, reserving 1 cup of the pasta water.
Add the pasta and ¾ cup pasta water to the skillet along with the chicken, lemon zest and juice and the remaining 4 tablespoons butter.
Return the skillet to medium-low heat and gently stir the pasta until the butter is melted, adding the remaining ¼ pasta water if the pasta seems too dry.
Remove the skillet from the heat, sprinkle with the grated cheese and parsley and toss before serving.
