Cook the pasta according to the package directions. Drain and set aside.
Pat the chicken breasts dry with a paper towel and rub with Cajun seasoning.
Heat the olive oil in a large skillet or cast-iron pan over medium-high heat. Add the chicken and cook for 5 to 6 minutes. Flip and cook for an additional 6 to 7 minutes, or until cooked through (internal temperature of 165°F). Transfer the chicken to a plate and cover with foil.
Melt the butter in the same skillet. Add the diced onion and sauté for 3 to 4 minutes. Add the minced garlic and cook until fragrant, about 1 to 2 minutes. Add the heavy cream, lemon zest, and grated Parmesan and whisk until the sauce has thickened, about 2 minutes. Season with salt and pepper, to taste.
Slice the chicken into thin strips.
Toss the pasta with the sauce and place it on a serving plate. Arrange the chicken strips over the pasta. Garnish with diced tomatoes, parsley, green onions, and shredded Parmesan. Lightly sprinkle with Cajun seasoning if desired. Serve immediately.
Slow Cooker
Season the chicken with Cajun seasoning.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and quickly sear on both sides. The chicken should not be cooked through at this point.
Place the seared chicken, butter, onions, garlic, heavy cream, and lemon zest in a slow cooker. Stir. Cover and cook on high for 4 hours or on low for 6 to 8 hours.
Separately, cook the pasta according to the package directions. Drain.
Serve the chicken and sauce over the pasta and top with diced tomatoes, parsley, green onions, and shredded Parmesan.