Season chicken breasts with Cajun seasoning, rubbing the seasoning on both sides.
Heat a large skillet over medium high heat and add extra virgin olive oil and 2 tablespoons of unsalted butter to the skillet.
Cook chicken (in batches if needed), about 3 to 5 minutes per side until cooked through (until it reaches an internal temperature of 165°F). Remove chicken from pan and set aside.
Bring a large, generously salted, pot of water to a boil. Cook linguini pasta according to package directions to al dente while preparing the sauce.
Reduce the heat under the skillet to medium and add the remaining tablespoon of butter, onions and bell pepper. Cook until the onions have softened, stirring frequently, about 5 minutes.
Stir in tomato paste until incorporated, cook until fragrant and paste becomes a deep, rusty red color, 2 to 3 minutes.
Stir in minced garlic and cook until fragrant, about 1 minute.
Add chicken broth, stirring until fully incorporated, and scraping any bits from the skillet. Cook for 3 to 4 minutes to reduce slightly.
Stir in the heavy cream and Parmesan cheese and continue stirring until the cheese has completely melted and sauce begins to thicken, 3 to 4 minutes.
Toss cooked pasta with sauce, adding pasta water as needed for a creamy consistency, ¼ to ½ cup. Season with salt and pepper to taste.
Slice cooked chicken and serve with pasta or stir into pasta if desired. Garnish with diced tomato, green onion and additional Parmesan cheese if desired.