Bring a large pot of salted water to a rolling boil. Add the fettuccine or linguine and cook according to the package instructions until al dente. Once cooked, drain the pasta in a colander, making sure to reserve 1/2 cup of the starchy pasta water for later.
Place the peeled and deveined shrimp in a medium bowl. Sprinkle the Cajun seasoning over them and toss well to ensure each shrimp is evenly coated in the spices.
Heat the olive oil in a large skillet over medium heat. Carefully add the seasoned shrimp to the skillet and sauté for about 2-3 minutes on each side, or until they turn pink and opaque. Once cooked, remove the shrimp from the skillet and set them aside on a plate.
In the same skillet, introduce the diced onion. Sauté for about 3-4 minutes until the onion becomes soft and translucent. Add the minced garlic and halved cherry tomatoes, continuing to cook for an additional 2 minutes until the tomatoes begin to soften and release their juices.
Slowly pour in the heavy cream and chicken broth, stirring well to combine the mixture. Raise the heat slightly and allow the sauce to simmer for 2-3 minutes, letting it thicken a bit. Season with salt and freshly ground pepper to taste.
Return the cooked shrimp to the skillet along with the fresh spinach and the drained pasta. Gently toss everything together to coat in the creamy sauce. If the sauce is too thick, gradually add some of the reserved pasta water until you reach your desired consistency.
Once everything is heated through and well mixed, remove the skillet from the heat. Serve the pasta immediately, garnished with a sprinkle of fresh parsley and optional grated Parmesan cheese on top.