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, combine the olive oil, chicken, Cajun seasoning, 2 tablespoons grated Parmesan, and a generous pinch of black pepper; toss to coat evenly.
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 the chicken to a plate.
To the same skillet, 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, whisk until smooth, and bring to a gentle simmer; cook about 5 minutes until slightly thickened.
Stir in 1 1/2 cups grated Parmesan until melted, then add the drained pasta and toss to coat. Cook together 2 minutes to marry the sauce and pasta.
Remove from heat, stir in the chopped parsley, slice the chicken, and serve the pasta topped with the sliced chicken and additional parsley if desired.
