Bring a large pot of salted water to a boil. Add 1 pound dried penne pasta and cook according to package directions for 1 minute less than al dente. Reserve 2 cups of the pasta cooking water, then drain. Meanwhile, make the sauce while the pasta is cooking.
Place 2 pounds chopped boneless, skinless chicken breasts in a medium bowl. Drizzle with 1 tablespoon of the olive oil, season with 1 tablespoon of the Cajun seasoning, and toss to coat. Heat a medium Dutch oven or heavy-bottomed pot over medium-high heat. Add the chicken in a single layer and cook undisturbed until deeply golden-brown on bottom, about 3 minutes. Flip the chicken and cook until cooked through, 3 to 5 minutes more. Transfer to a plate.
Add remaining 1 tablespoon olive oil to the Dutch oven. Add 1 finely chopped medium yellow onion, 2 thinly sliced medium bell peppers, 4 thinly sliced garlic cloves, and the remaining 1 teaspoon or 2 tablespoons Cajun seasoning (depending on the saltiness of the seasoning blend). Cook, stirring occasionally, until the onions are translucent and the peppers are crisp-tender, about 5 minutes.
Reduce the heat to medium-low. Add 1 (14.5-ounce) can fire-roasted diced tomatoes and their juices, 6 ounces cubed cream cheese, and 3 ounces finely grated Parmesan cheese. Cook, stirring constantly, until the cream cheese is fully melted, about 1 minute.
Return the chicken and any accumulated juices on the plate to the Dutch oven. Add 1 cup of the reserved pasta water and 2 teaspoons sherry vinegar, and stir to evenly distribute.
Add the pasta and cook, tossing constantly, until the sauce clings to the pasta, about 1 minute. Add more pasta water a tablespoon at a time as needed if the sauce is too thick. Taste and season with kosher salt and black pepper as needed. Garnish with more grated Parmesan cheese and finely chopped fresh parsley leaves if desired.