Sprinkle the blackening seasoning evenly over both sides of the chicken breasts.
Heat the olive oil in a cast iron skillet over medium-high heat until shimmering.
Add the seasoned chicken and cook about 5 minutes per side, or until cooked through; transfer chicken to a plate and keep warm.
Reduce heat to medium, add the butter to the skillet, then sauté the minced garlic and sliced red bell pepper about 2 minutes, until the pepper begins to soften.
Pour in the heavy cream and bring to a gentle simmer; cook for about 5 minutes to slightly thicken.
Lower the heat to low, then stir in the room-temperature cream cheese and 1 1/2 cups grated Parmesan until fully melted and the sauce is smooth.
Add the salt, black pepper, white pepper, and dried thyme to the sauce and stir to combine.
Toss the cooked, drained fettuccine into the sauce until evenly coated.
Divide the fettuccine among four plates, top each with a chicken breast, and sprinkle additional grated Parmesan to serve.