Sprinkle 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 chicken and cook about 5 minutes per side, or until cooked through; remove from pan and keep warm.
Reduce heat to medium, add the butter, then sauté the minced garlic and thinly sliced red bell pepper until the pepper begins to soften, about 2 minutes.
Pour in the heavy cream and bring to a gentle simmer; cook for about 5 minutes.
Lower the heat and add the room-temperature cream cheese, grated Parmesan, salt, black pepper, white pepper, and dried thyme; stir until the cheeses are melted and the sauce is smooth.
Add the cooked, drained fettuccine to the skillet and toss to coat thoroughly with the sauce.
Divide the fettuccine among 4 plates, top each with a blackened chicken breast, and sprinkle with additional Parmesan to serve.