Bring a large pot of salted water to a boil and cook the pasta until al dente according to package directions; drain and set aside.
In a bowl, toss the chicken with the olive oil, Cajun seasoning, 2 tablespoons grated Parmesan, and a generous pinch of black pepper until evenly coated.
Heat 2 tablespoons of the butter in a large skillet over medium-high heat. Add the chicken and sear until deeply golden and cooked through, about 5 minutes per side; turn off the heat and pour the lemon juice over the chicken, then transfer it to a plate.
To the same skillet, add the remaining 2 tablespoons butter, garlic, and crushed red pepper flakes and cook for about 1 minute until fragrant.
Pour in the chicken broth and heavy cream, whisking to combine. Bring the sauce to a gentle simmer and cook about 5 minutes until slightly thickened.
Stir in 1 1/2 cups grated Parmesan until melted and smooth, then add the cooked pasta and toss to coat. Cook together for 2 minutes to marry the flavors.
Remove the skillet from heat, stir in the chopped parsley, and season with salt and black pepper to taste.
Slice the seared chicken and serve it over the pasta, garnishing with additional parsley and the reserved 2 tablespoons Parmesan if desired.