Cook the Pasta: Bring a large pot of water to a boil and add 1 Tbsp salt. Cook the linguine pasta according to the package instructions until al dente. Reserve 1/2 cup of the pasta water, then drain the pasta, cover, and keep warm.
Prepare the Chicken: Pound the chicken breasts to an even thickness. Season all over with 1 1/2 Tbsp of cajun seasoning.
Cook the Chicken: In a large non-reactive skillet, heat 2 tsp of olive oil over medium-high heat. Once hot, add the seasoned chicken and sear on both sides. Reduce the heat to low and sauté until the chicken is cooked through and registers 165°F on a thermometer. Transfer the chicken to a cutting board, slice it into thin strips, and cover to keep warm.
Make the Sauce: In the same skillet, melt the butter over medium heat. Add the minced garlic and sauté for 30-60 seconds until fragrant. Add the diced tomatoes and sauté for another 2 minutes. Pour in the heavy whipping cream, remaining cajun seasoning to taste, and parmesan cheese. Stir and bring the sauce to a simmer. Adjust seasoning if necessary.
Combine and Serve: Add the sliced chicken and cooked pasta to the sauce. Toss until the pasta and chicken are well coated and warmed through. If the sauce is too thick, add the reserved pasta water a little at a time to reach your desired consistency. Serve the pasta with freshly grated parmesan and chopped parsley on top.