Pat the 4 boneless skinless chicken breasts dry and pound to even thickness. Sprinkle 2 tablespoons blackening seasoning evenly over both sides of the chicken.
Heat 2 tablespoons olive oil in a cast-iron skillet over medium-high heat until shimmering. Add the chicken and cook about 5 minutes per side, or until cooked through (internal temperature reaches 165°F). Remove the chicken from the skillet and keep warm.
Reduce heat to medium and add 2 tablespoons butter to the same skillet. When the butter has melted, add 2 minced garlic cloves and 1/2 red bell pepper (thinly sliced). Cook, stirring occasionally, about 2 minutes, until the pepper just starts to soften.
Pour in 1 1/2 cups heavy cream and bring to a gentle simmer. Simmer for 5 minutes, stirring occasionally.
Turn the heat to low. Add 4 ounces room-temperature cream cheese (cut into chunks to help melt), 1 1/2 cups freshly grated Parmesan cheese, 1/4 teaspoon salt, 1/2 teaspoon black pepper, 1/2 teaspoon ground white pepper, and 1/4 teaspoon dried thyme. Stir constantly until the cream cheese and Parmesan are fully melted and the sauce is smooth.
Add the cooked and drained (16-ounce) fettuccine to the skillet. Toss or stir the pasta in the sauce until it is evenly coated and heated through, about 1–2 minutes.
Slice the cooked chicken breasts into strips.
Divide the sauced fettuccine between 4 plates, top each portion with the sliced chicken, and sprinkle additional Parmesan cheese for serving.