Bring a large pot of salted water to a boil and cook the linguine or fettuccine until al dente according to package instructions; drain and set aside.
In a mixing bowl, combine the olive oil, chicken cutlets (or tenders), Cajun seasoning, 2 tablespoons grated Parmesan, and a generous pinch of black pepper; toss to coat evenly.
Heat 2 tablespoons butter in a large skillet over medium-high heat. Add the seasoned 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 the chicken to a plate.
Wipe the pan if needed, then return it to medium heat and add the remaining 2 tablespoons butter, chopped garlic, and crushed red pepper flakes; cook about 1 minute until fragrant.
Pour in the chicken broth and heavy cream, whisking to combine. Bring to a gentle simmer and cook about 5 minutes until the sauce slightly thickens.
Stir in 1 1/2 cups grated Parmesan until melted and smooth, then add the cooked pasta and toss in the sauce for about 2 minutes so the pasta is well coated.
Remove the pan from the heat, stir in the chopped parsley, slice the seared chicken, and serve the pasta topped with the sliced chicken and remaining Parmesan if desired.