Bring a large pot of salted water to a boil. Cook the pasta al dente, according to package directions. Drain.In bowl, toss together the olive oil, chicken, cajun seasoning, 2 tablespoons parmesan, and a big pinch of pepper. Melt 2 tablespoons butter in a large pan set over medium-high heat, add the chicken and sear until deeply golden on both sides, 5 minutes per side. Turn the heat off and pour over the lemon juice. Transfer to a plate.To the same pan, add 2 tablespoons butter, the garlic, and chili flakes. Cook 1 minute, until fragrant, then pour in the broth and milk. Whisk until smooth. Bring the sauce to a gentle simmer and cook 5 minutes, until slightly thickened. Stir in 1 1/2 cups parmesan, then toss in the pasta. Cook 2 minutes. Remove from the heat and stir in the parsley. Serve immediately topped with sliced chicken and parsley.