Cook the pasta in salted water according to the package instructions until al dente. Reserve about ½ cup of the pasta water.
Add the chicken to a medium mixing bowl and season with 2 teaspoons of Cajun seasoning. Toss the chicken to coat it in the seasoning and set the bowl aside.
Add the shrimp to a small mixing bowl and season with 1 teaspoon of Cajun seasoning. Toss the shrimp to coat with the seasoning and set them aside.
Heat the butter and olive oil in a deep heavy-bottomed skillet over medium-high heat. Cook the shrimp in the oil until just opaque and almost cooked through. Place the shrimp on a plate and set them aside.
Add the chicken to the pan and cook it for 3 to 5 minutes until almost cooked through. Add the bell pepper and onion and cook the vegetables until they soften. Add the garlic and cook for an additional minute or until the garlic is fragrant.
Add the tomatoes and chicken broth to the pan and give everything a good stir with a wooden spoon, scraping up any brown bits on the bottom of the pan. Add the remaining Cajun seasoning to the pan and season to taste with salt and pepper.
Continue to cook the chicken and vegetables for 3 to 5 minutes until the chicken is cooked and the liquid reduces a bit. Toss the shrimp in with the rest of the ingredients and cook until the shrimp are cooked through.
Add about half of the pasta and 2 tablespoons of pasta water to the pan and toss it with the rest of the ingredients until everything is combined. Add more pasta until until you are pleased with the pasta to sauce ratio. I had about ¼ of my pasta leftover. Allow the pasta to sit for about 10 minutes. The sauce will thicken as it sits. Add more pasta water if you want to thicken the sauce a bit more.