Place your chicken breast halves on a clean surface or plate. Generously coat both sides of each piece with the Cajun blackening seasoning. Use your fingers to press the spices into the meat so they stick during the cooking process.
Heat olive oil in a large heavy skillet over medium-high heat. Once the pan is shimmering, add the chicken. Cook for 5 to 7 minutes per side until the exterior is dark and charred and the internal temperature reaches 165°F. Remove the chicken from the pan and let it rest for 5 minutes before slicing it into strips.
While the chicken is cooking, bring a large pot of salted water to a boil. Add the fettuccine and cook according to the package instructions until it is al dente. Reserve about half a cup of pasta water before draining the noodles.
In a medium saucepan or the skillet used for the chicken (wiped out), melt the butter over medium heat. Add the minced garlic and sauté for 1 minute until fragrant. Pour in the heavy cream and bring to a gentle simmer. Whisk in the Parmesan cheese slowly until the sauce is smooth and thickened.
Add the drained fettuccine to the sauce. Toss well to coat every strand of pasta. If the sauce is too thick, add a splash of the reserved pasta water. Season with black pepper as needed.
Divide the creamy pasta into four bowls. Top each serving with a generous portion of the sliced blackened chicken. Garnish with fresh parsley and extra Parmesan cheese if desired.